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LEAVITT PATH ALGEBRAS AS FLAT BIMORPHICLOCALIZATIONS

P. N. ANH AND M. F. SIDDOWAY

Abstract. Refining an idea of Rosenmann and Rosset we show that the now widelystudied classical Leavitt algebra LK(1, n) over a field K is a ring of right quotientsof the unital free associative algebra of rank n with respect to the perfect Gabrieltopology defined by powers of an ideal of codimension 1, providing a conceptual,variable-free description of LK(1, n). This result puts Leavitt (path) algebras on thefrontier of important research areas in localization theory, free ideal rings and theirautomorphism groups, quiver algebras and graph operator algebras. As applicationsone obtains a short, transparent proof for the module type (1, n) (n ≥ 2) of Leavittalgebra LK(1, n) (n ≥ 2), and the fact that Leavitt path algebras of finite graphs arerings of quotients of corresponding ordinary quiver algebras with respect to the perfectGabriel topology defined by powers of the ideal generated by all arrows and sinks. Inparticular, the Jacobson algebra of one-sided inverses, that is, the Toeplitz algebra,can also be realized as a flat ring of quotients, further illuminating the rich structureof these beautiful, useful algebras.

1. Introduction

W. G. Leavitt introduced the extraordinarily insightful notion of ”module type”, animportant invariant of rings, in the late fifties of the last century. He showed that aunital ring is either a ring with IBN, that is, every free module has a unique rank,or is of module type (m,n)(1 ≤ m < n) where m,n are the smallest integers withrespect to the property that the free modules generated by a basis having m and nelements, respectively, are isomorphic. He [17] constructed (for the sake of simplicity) auniversal algebra of module type (1, n)(1 < n) over the field F2 of 2 elements. There areseveral proofs for this result provided by Cohn [4], Corner [8], [9] (together with anotherthat is unpublished, as far as we know) and for C∗-algebras by Cuntz [10]. Analogousproblems for Boolean algebras are discussed in Givent and Halmos’ book [14], Chapters

Date: August 30, 2021.2010 Mathematics Subject Classification. Primary 16S88, secondary 16P50.Key words and phrases. Leavitt path algebra, essential submodule, finitely generated module, local-

ization, flat epimorphism.The first author was partially supported by National Research, Development and Innovation Office

NKHIH K119934 and K132951, by both Vietnam Institute for Advanced Study in Mathematics (VI-ASM) and Vietnamese Institute of Mathematics. The second author is supported by a generous grantfrom the Colorado College Natural Sciences Division.

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http://arxiv.org/abs/2108.11987v1

2 P. N. ANH AND M. F. SIDDOWAY

27 and 45. Independently of Leavitt’s work, Cuntz [10] invented the twin C∗-algebraOn(n > 1) which has dominated research in operator algebras in the last half century.The intensive research in operator algebras initiated by Cuntz’s fundamental result ledfirst to operator graph algebras and then inspired ring theorists to consider ”Leavittpath algebras of directed graphs” (for a good account see [1]) which has become anactive subject in ring theory. The motivation for our work is the telling observation byRosenmann and Rosset [25] that the module type of the fc-localization of the free unitalassociative algebra of rank n (n ≥ 2) is (1, n). Namely, we show on one side that thecanonical inclusion of a free associative algebra A of rank n(> 0) over an arbitrary fieldK, that is, the quiver algebra KE of a graph E consisting of one vertex and n loops, intothe associated Leavitt path algebra LK(E), denoted as LK(1, n) is a flat bimorphism inthe category of K-algebras. On the other side, we construct precisely classical Leavittalgebras LK(1, n) (n > 0) as flat rings of right quotients of free associative algebras withrespect to a perfect Gabriel (two-sided ideal) topology defined by powers I l (l ∈ N) ofan ideal I ⊳ A of codimension 1, providing a conceptual, variable-free description ofLK(1, n). Moreover, we show that the two topologies, i.e., the ideal topology definedby the powers of an ideal I ⊳ A of codimension 1 and the one defined in Theorem2.1(b) with respect to a corresponding flat bimorphism from A into the associatedquotient ring which is isomorphic to the Leavitt algebra LK(1, n)(n > 1), coincide.This refined observation connects Leavitt (path) algebras to some central parts of otherrelated research, like quiver algebras of (finite) directed graphs, localization theory,free ideal rings and epimorphisms of rings. It shows also the naturality of Cuntz-Krieger conditions (CK1) and (CK2) in the definition of both Leavitt and operator graphalgebras. As further consequences, one obtains a new proof for some basic properties ofLeavitt algebras and we show that the Leavitt path algebra of a finite directed graphis a perfect localization of the quiver algebra with respect to the Gabriel topologyconsisting of certain well-defined finitely generated essential right ideals. Consequently,a Leavitt path algebra of a finite directed graph is flatly bimorphic to the ordinary quiveralgebra. It is also worth remarking that the determination of the module type of non-IBN, projective-free rings is frequently equivalent to the computation of an associatedGrothendieck group, emphasizing the natural importance of K-theory in the study ofLeavitt path algebras. Therefore, Leavitt’s notion of ”module type” (m,n) seems tobe a unifying idea that connects different algebraic realms like ring theory, K-theory,operator algebras and localization theory.

As a result, Leavitt path algebras can be viewed from two different vantage points.They are good examples for (until now nonstandard, but) natural localizations, andthus the techniques of localization, and results and methods from quiver algebras can beapplied in the study of Leavitt path algebras. Cohn’s localization by inverting matricesmakes it transparent that a canonical imbedding of the free associative algebra on aset X = (xij)(i = 1, · · · , m; j = 1, · · ·n, ; 1 < m < n) of free generators xij into theuniversal Leavitt algebra of module type (m,n), that is, the finitely presented algebra

LEAVITT PATH ALGEBRAS AS FLAT BIMORPHIC LOCALIZATIONS 3

generated by generators X = (xij), Y = (yji) subject to XY = 1m; Y X = 1n, is abimorphism; that is, both a monomorphism and an epimorphism, in the category ofalgebras. However, we do not know whether this bimorphism is flat and so the relationto Gabriel’s localization awaits further clarification. We also note that Bergman [2], [3]extended Cohn’s idea and provided yet another context for Leavitt’s results in a moregeneral setting. He also observed a connection between his universal constructionsand localizations but did not realize the striking fact that Leavitt algebras are (flatbimorphic) localizations!

2. Preminilaries: localization, digraphs and their algebras

A word about terminology. All fields are commutative. All algebras, modules areunital and associative over a field unless stated otherwise. Ideals and modules areconsidered with respect to algebras, that is, they are also at the same time vector spacesover a field. Finitely presented modules are factors of finitely generated modules byfinitely generated submodules. A(1−x) denotes always the left ideal {r−rx | r ∈ A} foran element x of a ring A and similarly for the right ideal (1−x)A. For further undefinednotions for rings, localizations or for Leavitt path algebras we refer to monographs [26]and [1], respectively.

The theory of Rings of quotients was introduced independently by Findlay and Lam-bek [12] and Utumi [27] in the late 1950’s. Utumi’s work is definitely important for usein Leavitt path algebras by permitting rings without identity. Namely, a ring Q (notnecessarily with identity) is a ring of (right) quotients of a subring R, and in this caseR is called dense (on the right) in Q if for any two elements q1, q2 ∈ Q, q1 6= 0, thereis r ∈ R such that q1r 6= 0, q2r ∈ R. It is trivial that the left annihilator of a densesubring is precisely the zero ideal, and dense right ideals are essential. Alternative ap-proaches to localization via torsion theory, calculus of fractions and quotient categoriesby localizing Serre subcategories were developed later in the sixties by Gabriel, Lambek,Morita, etc. For details we refer to books [11], [23], [26] and papers by Lambek [16],Morita [19], [20], [21], [22]. These citations are primarily meant to direct the reader toessential developments of the theory and to the rich collection of references included bythe authors.

Recall that a ring homomorphism φ : A → B is called an epimorphism if for anyring C and ring homomorphisms α, β : B → C, αφ = βφ implies α = β. Dually, aring homomorphism φ : A→ B is a monomorphism if for any ring C and any two ringhomomorphisms α, β : C → A, an equality φα = φβ implies α = β. Ring epimorphismsare not necessarily surjective but ring monomorphisms are always injective. Namely, ifthere is 0 6= a ∈ A with φ(a) = 0, then α, β : Z[X ] −→ A defined by putting α(X) = 0and β(X) = a, respectively, are two different ring homomorphisms from Z[X ] to Asatisfying φα = φβ. Two rings are bimorphic if there is a ring homomorphism betweenthem which is both a monomorphism and an epimorphism. For example, the ring Z ofintegers and the field Q of rationals are bimorphic. An epimorphism φ : A→ B is flat


if AB is a flat left A-module. In this case B is called a perfect right localization or aflat epimorphic right ring of quotients of A. For the sake of completeness and becauseof its importance in our study, we quote here a characterization of flat epimorphismswhich has been proved by a numbers of authors (for example, Findlay, Knight, Lazard,Morita, Popescu and Spircu, etc.), using various methods independently almost at thesame time.

Theorem 2.1 (Theorem XI.2.1 [26]). Let φ : A → B be a ring homomorphism. Thefollowing assertions are equivalent:

(a) φ is an epimorphism and makes B into a flat left A-module.(b) The family F of right ideals a of A such that φ(a)B = B is a Gabriel topology,

and there is a ring isomorphism σ : B → AF such that σφ : A → AF is the canonicalhomomorphism

(c) The following two conditions are satisfied:(i) For every b ∈ B there exist s1, · · · , sn ∈ A and b1, · · · , bn ∈ B such that bφ(si) ∈

φ(A) andn∑

i=1

φ(si)bi = 1.

(ii) If φ(a) = 0, then there exist s1, · · · , sn ∈ A and b1, · · · , bn ∈ B such that asi = 0

andn∑

i=1

φ(si)bi = 1.

Note that assertion (b) simply means that B is a quotient ring of A with respect tothe Gabriel topology F, and F is the finest among the Gabriel topologies T of A suchthat the ring AT of right quotients of A with respect to T is isomorphic to B. Moreover,for our aim it is worth noting that the verification of the implication (c) ⇒ (a) iselementary, and does not require any knowledge from localization theory. Condition (i)of assertion (c) has an important consequence in our study.

Corollary 2.2 (Exercise 11 of IV.4.16 [23] p.269). If φ : A→ B is a flat epimorphism,then every right ideal b of B satisfies b = φ(φ−1(b))B. Therefore the associated Gabrieltopology admits a basis consisting of finitely generated ideals which are also dense whenceessential provided that φ is also one-to-one, i.e., φ is a flat bimorphism. In particular,if φ is one-to-one, then b = (b ∩ A)B holds by identifying φ(a) with a for every a ∈ A.

Proof. The claim follows immediately from the observation that for any b ∈ B andsi ∈ A, bi ∈ B as in (i) of assertion (c), one has

b = bn

∑

i=1

φ(si)bi =n

∑

i=1

(bφ(si))bi.

The above equation shows also that B is therefore obtained by a kind of generalizedcalculus of fractions. The other claims are obvious in view of the fact that a right ideal∑

siA generated by the si is open by assertion (b) of Theorem 2.1, and the equalityn∑

i=1

φ(si)bi = 1. �


If φ : A → Q is a flat bimorphism, then Q becomes a subring of the maximal ringQmax(A) of right quotients of A by identifying each a ∈ A with φ(a) ∈ Q. It is well-known that Qmax(A) has a largest subring Qtot(A) (called the maximal flat epimorphicring of right quotients of A) which is a flat epimorphism of A and contains all flatbimorphisms of A. It is also well-known, and in fact, not hard to see that every elementq of Qmax(A) uniquely determines a largest right ideal dom(q) = {a ∈ A | qa ∈ A} ofA, called a maximal right ideal of definition of q, and q can be identified with an A-homomorphism q : dom(q) −→ A : a 7→ qa = q(a). This observation greatly simplifiesnotations when working inside maximal rings of quotients.

Since terminology in graph theory is notoriously nonstandard, we fix necessary nota-tions and concepts to help avoid confusion. We then remind the reader of the definitionsof both quiver and Leavitt path algebras, and make some germane observations aboutthese algebras which will be useful in the sequel.

A directed graph (or simply digraph) is a quadruple E = (E0, E1, s, r) of a non-emptyvertex set E0, an arbitrary arrow set E1, and source and range functions s, r : E1 → E0.A vertex v ∈ E0 is singular if it is either a sink or an infinite emitter according to|s−1(v)| = 0 or |s−1(v)| = ∞, respectively. v is a regular vertex if it is not singular.A finite path α of length n = |α| > 0 from a source s(α) = v0 to a range r(α) = vn

is a sequence α =v0•

a1−→v1• −→ · · ·

vn−1

•an−→

vn• , written as a finite word α = a1 · · · an,

of n arrows ai ∈ E1 satisfying s(ai+1) = vi = r(ai) (i = 1, · · · , n − 1). Moreover,a1 · · ·ai (0 ≤ i ≤ n) is called a head hα(i) (hα(0) = v0) of length i while ai+1 · · · an iscalled a tail tα(i) (tα(n) = vn) of colength i of α, respectively. Sometimes, for the sake ofsimplicity, we will omit the index α in functions h, t when the meaning is clear. Everyvertex v ∈ E0 is, by convention, a path of length 0 from v to v. The set of all finitepaths is denoted by F (E).

A useful device in applications of digraphs is the composition of paths. Namely, ifα = a1 · · · an and β = b1 · · · bm are paths of length n,m ≥ 0, respectively, satisfyingr(α) = s(β), then the composition or the product αβ is well-defined by concatenation

αβ = a1 · · · anb1 · · · bm.

The addition law of length |αβ| = |α|+ |β| holds obviously when αβ is well-defined.Reversing arrows gives the dual digraph E∗ of E. E∗ has the same vertex set E0 as

E, and the set {a∗|a ∈ E1} of arrows with s(a∗) = r(a), r(a∗) = s(a). Hence paths inE∗ are exactly α∗ = a∗n · · · a

∗1 for α = a1 · · · an ∈ F (E). Therefore the set of finite paths

in E∗ is denoted by F ∗(E) not by F (E∗) as is expected by convention, emphasizing thefact that ∗ is an anti-isomorphism between E and E∗.

Definition 2.1. The path algebra or the quiver algebra KG of a digraph G over a fieldK is the K-vector space KG with base F (G) such that a product µν (µ, ν ∈ F (G)) isthe composition of paths when r(µ) = s(ν), or 0 ∈ KG when r(µ) 6= s(ν).


Definition 2.2. The extended digraph of E, denoted by E, is the digraph constructedfrom E by adding all arrows a∗ (a ∈ E1). Paths in E lying in F (E) or in F ∗(E) arecalled real or ghost, respectively.

Definition 2.3. Let E be the extension of a digraph E = (E0, E1, s, r : E1 → E0) by

adding arrowsr(a)•

a∗

−→s(a)• for all a ∈ E1. The Leavitt path algebra LK(E) of a digraph

E over a field K is the factor of KE by the so-called Cuntz-Krieger relations

(1) (CK1) for any two arrows a, b ∈ E1 a∗b =

{

r(a) if a = b,

0 if a 6= b,

(2) (CK2) for every regular vertex v ∈ E0 v =∑

a∈s−1(v)

aa∗.

The Cohn path algebra CK(E) of E is the factor of KE by (CK1).

From the definition it is clear that KE and KE∗ are subalgebras of both CK(E) andLK(E). The set {αβ∗ | r(α) = r(β); α, β ∈ F (E)} is a K-basis of CK(E) but onlya set of generators for LK(E) over K in view of the Cuntz-Krieger condition (CK2).

However, there is, fortunately, no confusion when∑

kiαiβ∗i ∈ KE are used simply for

elements of both CK(E) and LK(E). By Cuntz-Krieger condition (CK1) the assignmentα ∈ F (E) 7→ α∗ ∈ F ∗(E) induces a standard, canonical involution in both LK(E) andCK(E) respectively.

Remarks 2.4.

(1) Cohn and Leavitt path algebras are in general factors of free associative K-algebras without identity, that is, of the algebras of polynomials in non-commutingvariables with zero constant term by ideals which are also K-spaces.

(2) These algebras are unital if and only if E0 is a finite set. Moreover, for digraphswith finite vertices they are finitely generated and they are finitely presented ifand only if the graphs are finite. Therefore for graphs of finite vertices Schreiertechniques, or in modern terminology, Grobner bases, are quite efficient tools intheir study as is nicely observed in Lewin’s work [18].

(3) Finite sums of vertices act as a set of local units for both Cohn and Leavitt pathalgebras.

3. Leavitt algebras L(1,n) are rings of quotients

Because of its importance, we devote this section to the particular case of Leavittalgebras, although one could incorporate this study into the next section. We refinefirst the nice observation by Rosenmann and Rosset [25] by showing that universalLeavitt algebras LK(1, n)(n > 0) of a graph consisting of one vertex together withn loops over an arbitrary field K are rings of quotients of the unital free algebras of


rank n with respect to the Gabriel topology of certain finitely generated essential rightideals. As an application we offer an alternative elementary and direct approach tobasic properties of Leavitt algebras without citing difficult deep results or quoting fromrelated results for Cuntz algebras On.

Fix an integer n > 0 and a commutative field K. By definition, LK(1, n) is a Leavittpath algebra of a digraph with one vertex and n arrows ai. Consequently, LK(1, n) is the

K algebra generated by ai, a∗i subject to relations

n∑

i=1

aia∗i = 1, a∗jai =

{

1 if j = i,

0 if j 6= i.

Elements in LK(1, n) are not necessarily unique linear combinationsm∑

i=1

αiβ∗i where αi

and βi are monomials in the ai, respectively, and ∗ is the involution of LK(1, n) induced

by sending ai to a∗i . Moreover, the relation a∗jai =

{

1 if j = i,

0 if j 6= iimplies that for

monomials α and β in the ai with |β| ≤ |α| the product β∗α is either 0 or a monomialin the ai, i.e., an element in the free algebra A = K〈a1, · · · , an〉. Note that LK(1, 1) isthe Laurent polynomial algebra K[x, x−1] which is obviously a ring with IBN. We nextverify

Theorem 3.1. The canonical inclusion of A into LK(1, n) is a flat epimorphism, i.e.,it is a flat bimorphism.

Proof. In view of Theorem 2.1 it is enough to verify its assertion (c). Since A is asubalgebra of LK(1, n), claim (ii) of (c) is obvious. The case of the Laurent polynomialalgebra K[x, x−1], i.e., the case of LK(1, 1), is trivial because it is the localization of thecommutative polynomial algebra K[x] with respect to the set {1, x, x2, · · · }. However,it is worth noting that the associated (perfect) Gabriel topology is given by the filterbase of ideals xlK[x] and so there are finite codimensional ideals which are not openwith respect to this topology! For the case of n > 1 claim (i) of (c) is fortunatelyan immediate consequence of the following two observations. First, a product β∗α isalways either 0 or a monomial in A if |α| ≥ |β| where α and β are monomials in theai’s. Secondly, one has

1 =n

∑

i=1

aia∗i = · · · =

∑

|αk|=|βk|=l

αkβ∗k =

∑

|αk|=|βk|=l

αk(n

∑

i=1

aia∗i )β

∗k =

∑

|αk|=|βk|=l+1

αkβ∗k

for every l ∈ N where αk, βk are monomials in the ai’s. �

From now on we assume n > 1. By assertion (b) of Theorem 2.1 a right idealR of the free associative algebra A = K〈a1, · · · , an〉 is open in the Gabriel topologyT determined by the canonical flat epimorphism A → LK(1, n) = AT if and only ifRLK(1, n) = LK(1, n). Consequently, there are finitely many elements ri ∈ R, ti ∈LK(1, n) with

∑

rit1 = 1 whence∑

riA ⊆ R is also open, hence∑

riA is open in the


topology T. Consequently,∑

riA is a finitely generated essential right ideal of A byCorollary 2.2. By (3.3) Theorem [25]

∑

riA is a right ideal of finite codimension ofA and so R is finite codimensional, too. Again by (3.3) Theorem [25] R is a finitelygenerated essential right ideal of A. This shows that a Gabriel topology T of A definingthe canonical flat epimorphism A → LK(1, n) consists of certain finitely generatedessential, i.e., finite-codimensional right ideals whence T is coarser than the fc-topologyinvented in [25] consisting of all finite codimensional right ideals of A. However, one hasa better, visual description of perfect localizations of A resulting in the Leavitt algebra

LK(1, n) as follows. The equalityn∑

i=1

aia∗i = 1 shows that the ideal I =

n∑

i=1

aiA is open

with respect to the topology T in view of (b) Theorem 2.1. Moreover I is clearly a freeright A module of rank n with canonical projections a∗i . More generally, the equalities1 =

∑

|αk|=|βk|=l

αkβ∗k (l ∈ N) where αk and βk are monomials in the ai’s imply that ideal

powers I l (l ∈ N) are open in T, and free right A-modules of rank nl with canonicalprojections β∗

k (|βk| = l). These facts show, in view of Proposition VI.6.10 [26], thatthe ideal topology F induced by powers I l is a Gabriel topology. Moreover by definitionthe ring of right quotients AF is obviously the Leavitt algebra LK(1, n). Therefore wehave proved the following result.

Theorem 3.2. Let A be the free associative algebra K〈a1, · · · , an〉 of rank n > 1 overthe field K where ai are free variables. The topology F defined by powers of the idealn∑

i=1

aiA is a perfect Gabriel topology and the ring of right quotients AF is the Leavitt

algebra LK(1, n). The topology F is obviously the coarsest Gabriel topology definingLK(1, n).

We shall see later in Corollary 3.8 that the ideal topology F defined in Theorem 3.2coincides with the topology T induced by the canonical flat bimorphism A→ LK(1, n).Moreover, the above argument shows also that an arbitrary basis {s1, · · · , sn} of the

free right A-module IA =n∑

i=1

aiA, defines the representation of the ring AF of right

quotients as the Leavitt algebra LK(1, n) with respect to injections (partial isometries)

si : A → siA : r ∈ A 7→ s1r and projections s∗i :n∑

i=1

siA → A if A = K〈s1, · · · , sn〉. It

is well-known and easy to verify that the si’s are algebraically independent over K butunclear whether these si generate A. In any case LK(1, n) = AF is isomorphic to itssubalgebra generated by si and s∗i (i = 1, 2, · · · , n) but this subalgebra need not coincidewith LK(1, n) in general. However, if si (i = 1, · · · , n) are images of the correspondingai’s under any K-algebra automorphism φ of A, then the Leavitt algebra LK(1, n) = AF

can be clearly generated by si, s∗i (i = 1, · · · , n). This reveals a close relation between

Leavitt algebras and automorphism groups of free associative algebras. More generally,any right ideal R of codimension 1 is a two-sided ideal of A, as is easily seen, and


furthermore is a free right A-module of rank n by the Schreier-Lewin formula [18].Writing ai = ri + ki for appropriate ri ∈ R, ki ∈ K (i = 1, · · · , n) one sees immediatelythat the ri generate A. We claim that the ri are algebraically independent over K.

Namely, a dependencen∑

i=1

riti = 0 (ti ∈ A) impliesn∑

i=1

aiti =n∑

i=1

kiti whence ti = 0 follows

for all indices i by the unique normal form a = k+n∑

i=1

aibi (k ∈ K; bi ∈ A) for every non-

zero element a of A. This result fits with the well-known fact that only the field K canbe recovered from its free unital associatve algebras, but free variables are not uniquelydetermined! Even the associated non-unital free associative algebras are not uniquelydetermined inside the unital free associative algebras. They correspond uniquely tomaximal (one-sided or two-sided) one-codimensional ideals of the considered unital freeassociative algebras. This result establishes a close connection between automorphismsof A and ideals of codimension 1 with an appropriate predescribed basis. Therefore weobtain a coordinate-free description of Leavitt algebras together with their close relationto automorphisms of free associative algebras.

Corollary 3.3. A ring AF of right quotients of a free associative algebra A of rank nover a field K with respect to the ideal topology defined by an ideal I of codimension1 is isomorphic to the Leavitt algebra LK(1, n). There is a one-to-one correspondencebetween representations of AF as a Leavitt algebra over K generated by a suitable basis{r1, · · · , rn} of I and automorphisms of A.

As applications we derive below some basic well-known properties of Leavitt algebrasLK(1, n). By using Cohn’s theory of free ideal rings which can be found succinctlyand transparently in the last section of [6], and repeating the insightful argument ofRosenmann and Rosset [25] on module type one, we can directly and easily re-obtainfundamental results on LK(1, n). The argument of Rosenmann and Rosset [25] onmodule type provides, in particular, a new conceptual way to determine the moduletype of certain rings.

Theorem 3.4. For n > 1 the Leavitt algebra LK(1, n) is simple and all of its right idealsare free, whence it is hereditary and all projective modules are free, too. Moreover, themodule type of LK(1, n) is (1, n) and its Grothendieck group K0(LK(1, n)) is isomorphicto the cyclic group of order n− 1.

Proof. Let A = K〈a1, · · · , an〉 (n > 1) be the free associative algebra over K generatedby free variables ai and consider LK(1, n) as the perfect localization AF of A with respect

to the ideal topology F defined by the idealn∑

i=1

aiA. If I is a non-zero ideal of LK(1, n),

then the ideal I ∩A of A is also non-zero by Corollary 2.2. Now we can use Cohn’s trickpresented in the proof of [4] Proposition 8.1, by considering an element 0 6= a ∈ I ∩ Asuch that a, as a non-commutative polynomial in variables ai, has a minimal number of


nonzero coefficients and also has a minimal total degree among such elements of I ∩A.By multiplying on the left with certain a∗i ’s and on the right with ai’s if necessary,one sees clearly that such an element a in I ∩ A must be a nonzero scalar, that is,I = LK(1, n) when LK(1, n) is simple.

Again by Corollary 2.2 we have J = (J∩A)LK(1, n) for every right ideal J of LK(1, n).Now, by Cohn’s result [6] A is a free ideal ring, whence J ∩ A is a free A-module, i.e.,J ∩ A =

∑

biA = ⊕biA for appropriate elements bi ∈ A satisfying bir = 0 ⇒ r = 0for each index i and every element r ∈ A. Consequently, by Corollary 2.2 we haveJ = (J ∩ A)LK(1, n) = (

∑

biA)LK(1, n) = (⊕biA)LK(1, n) = ⊕biLK(1, n) is a freeright LK(1, n)-module. Therefore, LK(1, n) is a hereditary algebra such that everyprojective module is free, i.e., a projective-free algebra.

To finish the proof we have to compute the module type of LK(1, n). We use here thestreamlined argument of Rosenmann and Rosset [25] for the module type of LK(1, n)with essential simplifications. To see that (1, n) is the module type of LK(1, n) it isclearly enough to show that if LK(1, n) is also free of rank m > 1, then n − 1 dividesm − 1. This means that there are elements bj ∈ LK(1, n) (j = 1, · · · , m) satisfyingLK(1, n) =

∑

bjLK(1, n) = ⊕bjLK(1, n) such that all right ideals bjLK(1, n) (j =1, · · · , m) are isomorphic to LK(1, n), i.e., multiplication by bj on the left is injective onLK(1, n). Therefore the right ideals bjdom(bj) ∼= dom(bj) (j = 1, · · · , m) of A are freeright A-modules of rank ≡ 1 mod(n− 1) by the finite codimensionality of dom(bj) andthe Schreier-Lewin formula [18]. This shows that R =

⊕

j bjdom(bj) is a free A-module

of rank ≡ m mod(n−1). On the other hand, R is an essential right ideal of A. Namely,

for any 0 6= a =m∑

i=1

bjqj ∈ A (qi ∈ AF = LK(1, n)) with at least one of the qj 6= 0, say

q1, there are nonzero elements r1 ∈ q1dom(q1)∩ (dom(b1)∩ (∩mj=1dom(qj))); r2, · · · , rm ∈

A such that qjr1r2 · · · rj ∈ dom(bj) for all j = 2, · · · , m in view of the fact that alldom(cj), dom(qj) are essential right ideals of A and q1dom(q1) 6= 0 by q1 6= 0. By achoice of r1, one has 0 6= q1(r1) = q1r1 ∈ dom(b1) whence 0 6= (b1q1(r1))r2 · · · rm =b1q1(r1r2 · · · rm) = b1q1r1 · · · rm ∈ b1dom(b1) because A is a domain. Similarly one getsa product such that ar1 · · · rm 6= 0 holds and all bjqjr1 · · · rm are not necessarily nonzeroelements of bjdom(bj), respectively, for all j > 1. Therefore R is both finitely generatedand an essential right ideal of A, whence R is a right ideal of A of finite codimensionby (3.3) Theorem [25]. Hence again by the Schreier-Lewin formula [18] R is a free rightA-module of free rank ≡ 1 mod(n− 1). Therefore m− 1 ≡ 0 mod(n− 1) holds, i.e., nis the smallest integer satisfying AF

∼= AmF , whence the module type of LK(1, n) = AF

is (1, n). Consequently, the Grothendieck group K0(LK(1, n)) is cyclic of order n − 1because projective modules over LK(1, n) are free, completing the proof. �

Theorem 3.4 essentially extends the class of projective-free rings. It shows also thatthe determination of module type is indeed equivalent to the computation of the associ-ated Grothendieck group for these particular rings. Typical examples for projective-free


rings are local rings, principal ideal domains, free associative algebras or more gener-ally free ideal rings and commutative polynomial algebras over fields of finitely manyvariables (Serre’s conjecture, now a theorem by Suslin and Quillen).

Remark 3.1. The case n = 1 is not very interesting but worth noting because it helpsus understand certain aspects of localization. The localization of K[x] with respect tothe Gabriel topology (consisting of all finite codimensional ideals) is the field K[x] ofrational functions, while K[x, x−1] as LK(1, 1) is the localization with respect to theGabriel topology of all ideals containing some powers of x. Since the maximal ringof quotients of K[x] is commutative, the Jacobson ring of one-sided inverses, i.e., theCohn algebra of the graph of one vertex together with one loop, cannot be a (perfect)localization of its quiver algebraK[x]. We shall discuss this situation in the next section.

We end this section with some results on certain localizations of free associativealgebras of rank n. Rosenmann and Rosset [25] showed that finite codimensional rightideals of a free associative algebra A constitute a Gabriel topology of A, called an fc-topology. By (3.3) Theorem [25], these right ideals are exactly the finitely generatedessential right ideals. Consequently, by [26] Proposition XI.3.3 and Proposition XI.3.4(d) one obtains that the fc-localization is a perfect localization. In particular, the

localization QfcK (n) is the field of rational functions in case n = 1. If φ : A → B

is an arbitrary flat bimorphism of A of rank n > 1, then any open right ideal J ofA with respect to the topology defined by φ must contain an open finitely generatedessential right ideal of A by Corollary 2.2 and so J contains an open right ideal of finitecodimension by (3.3) Theorem [25]. This implies that J is also finite codimensional,

i.e., J is open with respect to the fc-topology. This shows that QfcK (n) coincides with

the maximal, even largest (with respect to inclusion) flat epimorphic right ring Qtot(A)of quotients of A defined in Chapter XI.4 [26]. Therefore the same argument for Leavitt

algebras holds also for QfcK (n).

Theorem 3.5. The fc-localization QfcK (n) of a free associative algebra A of rank n over

a field K is a perfect localization, even coincides with the largest flat bimorphic rightring Qr

tot(A) of quotients of A. QfcK (n) is a simple, projective-free algebra such that its

module type is either IBN for n = 1 or (1, n) if n > 1. Furthermore, its Grothendieckgroup is a cyclic group of either infinite order for n = 1 or n− 1 for n > 1.

Proof. The only less trivial claim is the simplicity of QfcK (n). If n = 1, then the quotient

ring is a field of rational functions in one variable, hence it is simple. If n > 1, then thering of quotients with respect to fc-localization is simple by the same argument used inthe first paragraph proving Theorem 3.4. Namely, any nonzero ideal I of Qfc

K(n) has anonzero intersection with A and so its elements are linear combinations of monomials inthe ai and by definition the elements a∗i are also elements of Qfc

K(n), so this intersectionmust contain a non-zero constant, completing the proof. �


Proposition 3.6. The maximal ring of right quotients of a free associative algebra isa flat left module but it is not a perfect localization if the rank is not 1. This means,in this case, that the canonical embedding of the free associative algebra of rank > 1is not a flat epimorphism. In particular, the maximal rings of right quotients of freeassociative algebras are simple.

Proof. If the rank of the free associative algebra is not 1, then the commutator ideal isinfinite codimensional. Moreover, the commutator ideal, in fact, every nonzero ideal of afree associative algebra, is always an essential hence dense ideal because free associativealgebras are both domains and nonsingular rings. This shows by [25] (3.3) Theorem,and [26] Proposition XI.3.4(d) that the dense topology is not perfect. However, bySandomirski’s result [[26] Proposition XII.6.4] the maximal ring Qmax(A) of quotients isa flat left A-module. For the last claim one observes that the case of rank n = 1, that is,the case of polynomial algebra A of one variable, is trivial because the maximal ring ofquotients coincides with the rational function field. For the case of rank n ≥ 2, Qmax(A)contains infinitely many simple subalgebras isomorphic to LK(1, n), and referencing anyone of them implies the claim. This completes the proof. �

Remark 3.2. It is important to emphasize the following concerning the proof of Propo-sition 3.6. Namely, the commutator ideal of a free associative algebra of rank n ≥ 2is a finitely generated two-sided ideal generated by aiaj − ajai for all different pairs(i, j) but not a finitely generated one-sided ideal! Therefore it is not a finite codimen-sional one-sided ideal, but the ordinary commutative polynomial algebras are finitelypresented algebras. Moreover, finitely generated associative algebras are not necessarilynoetherian algebras!

Assume now n > 1 and consider an arbitrary flat bimorphism φ : A −→ Q of a freeassociative K-algebra A of rank n together with the associated Gabriel topology. If Jis an arbitrary open right ideal, then it is finitely generated and essential hence finitecodimensional by (3.3) Theorem [25]. Therefore the largest two-sided ideal containedin J which is exactly the annihilator ideal I = {a ∈ A |Aa ⊆ J} of A/J , is obviouslyopen, and finite codimensional. This implies that A/I is a finite-dimensional K-algebra.Moreover, if I is an arbitrary two-sided ideal of A such that A/I is a finite-dimensionalalgebra, then I as a right ideal of A is finitely generated by the Schreier-Lewin formula[18]. Furthermore, I is also a right essential ideal because A is a domain whence I isa right dense and finite-codimensional ideal. Consequently, there is a bijection betweenflat bimorphisms φ : A −→ Q of the free associative K-algebra A and sets Φfs ofmaximal two-sided finite-codimensional ideals I ⊳ A by [26] Proposition VI.6.10. Openright ideals of A are exactly the right ideals containing some products of ideals from Φfs

by [26] Proposition VI.6.10. Hence the fc-topology is a Hausdorff ideal topology andA is residually finite, pointing out a striking similarity with the structure of a profinitetopology for groups. In particular, flat bimorphisms of free associative algebras of rankn > 1 incorporate both ring theory and module theory of finite dimensional algebras


which are generated by at most n generators. However, the completion of A withrespect to the fc-topology is not the direct products

∏

I lim←−A/Il of inverse limits lim←−A/I

l

of the canonical inverse systems {A/I l | l ∈ N} where I runs over all maximal finite-codimensional ideals of A, because maximal two-sided finite co-dimensional ideals arenot commutable. As in the case of both Qfc(n)K and LK(1, n) (n ≥ 2), if there is anopen maximal one-codimensional ideal of A with respect to a finest topology defined bya flat bimorphism A −→ Q, then one obtains by the same proof for Qfc(n)K , that Q isa simple and right hereditary ring, projective right Q-modules are free, and the moduletype of Q is (1, n) whence K0(Q) ∼= Z/(n− 1)Z. In the general case of an arbitrary flatbimorphism A −→ Q all that we can say is that Q is right hereditary, right projective-free; i.e., right projective Q-modules are free. Therefore we have verified the followingresult.

Proposition 3.7. Let A be a unital free associative K-algebra of rank n ≥ 1. Then thefc-topology of A is a Hausdorff topology having a basis of finite-codimensional two-sidedideals. More generally, there is a bijection between flat bimorphisms φ : A −→ Q andsets Φfs of maximal two-sided finite-codimensional ideals I of A. Namely, a two-sidedideal I belongs to Φfs if and only if IQ = Q and I is maximal. A right ideal R isopen in the finest Gabriel topology defined by φ if and only if it contains a products ofideals from Φfs. Q is then a right hereditary, right projective-free ring. If Φfs containsa maximal one-codimensional ideal, then Q is a simple ring of module type (1, n).

As an immediate consequence of Proposition 3.7 we are now in position to showthat the ideal topology F defined by a maximal right ideal I of codimension 1 and thetopology T given by the corresponding flat bimorphism A→ AI

∼= LK(1, n) of the freeunital associative algebra A of rank n > 1 coincide.

Corollary 3.8. Let A be a free unital associative algebra of rank n > 1 over a fieldK and F an ideal topology defined by a maximal right ideal I of codimension 1. ThenF coincides with the toplogy T induced by the flat bimorphism A → AI

∼= LK(1, n).In particular, if I is an arbitrary finite codimensional two-sided ideal of A, then theideal topology F defined by powers of I coincides with the topology T determined by thecorresponding flat bimorphism A −→ AF.

Proof. To verify the statement, it is enough to see that a right ideal R of A containssome power of I if RAF = AF holds. Without loss of generality one can assume thatI is the ideal

∑n

i=1 aiA and so AT = AF = LK(1, n) in view of our standard notation.Assume indirectly that R is not open with respect to the topology F, then the two-sided ideal annA A/R satisfies annAA/RLK(1, n) = LK(1, n) as we have already seen.Consequently, there is a maximal two-sided ideal J 6= I that satisfies JLK(1, n) =LK(1, n). Since J is a free right A-module of finite rank and J is open with respectto the topology T, any projection q of J onto A is also an element in LK(1, n) whichis obviously not contained in A. Since q can be expressed as a linear combination


∑

kiαiβ∗i where αi, βi are monomials in the a1, · · · , an the domain of definition of q

contains some power I l for some l ∈ N. Therefore the domain of definiton of q containsJ + I l ⊇ (I + J)l = A, whence q ∈ A holds which is a contradiction! Hence T = F. Thelast assertion can be checked in the same manner. �

In view of Proposition 3.7 we are going to determine the module type of all flatbimorphisms φ : A → Q. By way of preparation, we consider the case when Q is thering of right quotients of A with respect to the I-adic topology where I is a maximalfinite-codimensional ideal of A.

Theorem 3.9. Let I be a maximal ideal of a free associative K-algebra A of rank n ≥ 2over a field K such that A/I is a matrix ring Dm over a division ring D of dimension lover K. If Q is a ring of right quotients of A with respect to the perfect Gabriel I-adictopology defined by the powers I l (l ∈ N), then the module type of Q is (1, lm(n−1)+1).

Proof. By the previous results of this section we already know that the canonical imbed-ding A → Q is a flat bimorphism. First we show that the I-adic topology coincideswith the topology defined by the flat bimorphism A → Q. For this purpose, it is clearlyenough to see that a right ideal R of A is open in the I-adic topology if RQ = Q holds.Namely, the equality RQ = Q implies ri ∈ R and qi ∈ Q with

∑

i r1qi = 1. There isthen a positive integer j that all qi are well-defined on Ij. Consequently, for all a ∈ Ij

we have a =∑

i riqi(a) =∑

i ri(qia) ∈ R whence R is open in the I-adic topology. IfR is the inverse image of a maximal right ideal of A/I ∼= Dn, then R has codimension|D : K|m = lm. Therefore R has a free generator set {r1, r2, · · · , rlm(n−1)+1 = rd} by theSchreier-Lewin formula [18]. For each index i ∈ {1, 2, · · · , d} let r∗i ∈ Q be defined on Rby sending ria ∈ R to a and all the other rj

′s (j 6= i) to 0. Then we have the identities

r∗j ri =

{

1 if j = i,

0 if j 6= iand

∑

rir∗i = 1 showing an isomorphism QQ

∼= QdQ. Therefore

to show that Q has a module type (1, d) = (1, lm(n−1)+1) it is enough to show that ifQQ∼= Qm

Q (m > 1), then lm(n−1) divides m−1. To this end, we observe first the follow-ing. If q is an arbitrary non-zero element of Q, then dom(q) is open in the I-adic topol-ogy, whence dom(q) contains some positive power I t (t ∈ N). Consequently, A/dom(q)can be considered as a module over a finite-dimensional primary K algebra A/I t whencethe K-dimension of A/dom(q), i.e., the K-codimension of dom(q), belongs to lmN. Infact, this remark holds for the codimension of any open right ideal in the I-adic topol-ogy. The isomorphism QQ

∼= QmQ implies the existence of m elements ci ∈ Q with trivial

right annihilator satisfying Q =m∑

i=1

ciQ = ⊕mi=1ciQ. By the previous remark we have

dom(ci)A ∼= cidom(ci) having codimension mlni with appropriate positive integer ni

for each index i ∈ {1, · · · , d}. Therefore the right ideal J =m∑

i=1

cidom(ci) = ⊕mi=1ciQ


is a free right A-module of rank m + (m∑

i=1

ni)ml(n − 1) by the Schreier-Lewin formula

[18]. Consider now an arbitrary non-zero element c ∈ Q. By Theorem 2.1 (c)(1)there are elements a1, · · · , at ∈ A; qi ∈ Q satisfying cai ∈ A,

∑

i aiqi = 1 whenceai ∈ dom(c) and c = c

∑

i aiqi =∑

i(cai)qi ∈ (cdom(c))Q hold. Consequently, one has

JQ = (m∑

i=1

cidom(ci))Q =m∑

i=1

(cidom(ci))Q =m∑

i=1

ciQ = Q implying that J is open in the

I-adic topology. Hence by the Schreier-Lewin formula, J has the rank tlm(n − 1) + 1for some positive integer t ∈ N. This shows that lm(n − 1) divides m − 1, completingthe proof. �

By considering inverse images of maximal ideals of the commutative polynomials overK in n variables one gets maximal two-sided finite-codimensional ideals of A. Sincethere are elements of A which are not necessarily (non-commutative) polynomials inri by comparing ranks, it seems possible that Q is not simple and that there does notexist a natural, canonical involution on Q! It is well-known and easy to see that theK-subalgera of A generated by the ri is a free associative algebra of rank lm(n− 1) + 1over K. Furthermore it is also important to look for a normal form for elements of aflat bimorphism A → Q when Q is a ring of right quotients of a free associative algebraof rank n ≥ 2 over a field K with respect to the I-adic topology and I is a maximalfinite-codimensional ideal of A such that A/I is a finite-dimensional division K-algebra.In this case, if IA = ⊕riA is a free module with respect to a basis ri (i = 1, 2, · · · , d),

then one can define the usual elements r∗i ∈ Q such that r∗j ri =

{

1 if j = i,

0 if j 6= iand

∑

rir∗i = 1 hold. It is clear that Q is a K-algebra generated by A and these r∗i ’s, but

it is a unknown whether elements of Q can be written in the form∑

ajb∗j with suitable

aj ∈ A and monomials b∗j among the r∗i ’s.We are now in position to determine the module type of an arbitrary flat bimorphism

A → Q as follows.

Theorem 3.10. Let A be a free unital associative algebra of rank n ≥ 2 over a field Kand A → Q an arbitrary flat bimorphism, i.e., there is a set Λ of maximal two-sidedfinite codimensional ideals of A such that Q is a ring of right quotients of A with respectto the Gabriel topology TΛ induced by products of ideals from Λ. For each ideal Iλ ∈ Λthe factor ring A/Iλ is a matrix ring Dmλ

over a division ring D of dimension lλ overK. Put dλ = lλmλ and let d be the greatest common divisor of the dλ’s. Then themodule type of Q is (1, d(n − 1) + 1). In addition, if the ideals of Λ are commutable,then the completion of A with respect to Tλ is the direct products

∏

I lim←−A/Il of inverse

limits lim←−A/Il of the canonical inverse systems {A/I l | l ∈ N} where I runs over all

elements of Λ.


Proof. One can see immediately from the proof of Theorem 3.9 that Q ∼= Qdλ(n−1)+1

as right Q-modules for each index λ ∈ Λ. Consequently, one gets QQ∼= Qd(n−1)+1 by

elementary number-theoretic reasoning because d is the greatest common divisor of thedλ’s. Therefore to show that d(n − 1) + 1 is the module type of Q it is enough toshow that d(n− 1) is a divisor of m− 1 provided QQ is isomorphic to Qm

Q . To see thisclaim we observe the following. If R is any open right ideal of A with respect to thetopology TΛ, then the annihilator ideal I of A/R is also finite-dimensional whence Ris a finite-codimensional A/I-module, whence the K-codimension of R is a sum of thedimensions of simple A/I-subfactors appearing in its composition series. Consequently,the K-codimension of R is a multiple of d. Now the same proof for Theorem 3.9 impliesthat d(n− 1) is a divisor of m− 1 when the module type of Q is (1, d(n− 1) + 1)

To finish the proof we observe the following. If I and J are two different commutablecoprime two-sided ideals, i.e., A = I + J, IJ = JI, of A then we have

A = (I + J)(I + J) = I + J2 = I2 + J = I2 + J2 = · · · = I l + Jm ∀l, m ∈ N,

and

I ∩ J = (I ∩ J)(I + J) = IJ + JI = IJ = JI.

By iterating these equalities one obtains the following more general equalities for finitelymany pairwise coprime, commutable ideals I1, · · · , Im

Ini

i + ∩j 6=iInj

j = A & ∩ Ini

i =∏

i

Ini

i ∀ni ∈ N.

Consequently, the topology TΛ is the same one induced by finite intersections of powersof ideals from Λ, and factors of A by open ideals are finite direct sums

∏

A/Ini

i∼=

A/∏

Ini

i = A/∩Ini

i with Ii ∈ Λ. This shows a topological isomorphism Q =∏

I lim←−A/Il

where Q is the completion of Q with respect to TΛ, because they are inverse limits ofthe same inverse system! This completes the proof. �

It is worthwhile to note that it is not a routine exercise to precisely write down the2(d(n − 1) + 1) elements xi, yi(= x∗

i ) (i = 1, · · · , d(n − 1) + 1) of Q in Theorem 3.10

satisfying the equalities yjxi =

{

1 if j = i,

0 if j 6= iand

∑

xiyi = 1 which imply that Q

has the module type (1, d(n− 1) + 1). It is also unclear whether the xi’s can be takenfrom A.

Since one can embed free associative algebras in division rings, there are localizationsof free associative algebras in Cohn’s sense (see also Lambek [16] and Morita [22])which are ring epimorphisms. These localizations are, however, no longer localizationsin Gabriel’s sense.


4. Leavitt path algebras of finite digraphs are flat bimorphisms

We primarily consider in this section Leavitt path algebras of finite digraphs. Themain aim of this section is to prove

Theorem 4.1. Let E be a finite digraph whence every vertex is either regular or a sink.Then the inclusion of the quiver algebra KE in the Leavitt path algebra LK(E) is aflat epimorphism. Consequently, LK(E) is the localization of KE with respect to theGabriel topology of all right ideals of KE which generate the right ideal LK(E)LK(E).

Proof. Since KE is a subalgebra of LK(E), we have only to verify condition (i) ofassertion (c) in Theorem 2.1. Although the idea of the proof is the same as for Theorem3.1 we have to refine the argument used there. Consider an arbitrary nonzero element

r ∈ LK(E). Write r =n∑

i=1

kiαiβ∗i (0 6= ki ∈ K) as a linear combination with possibly

the smallest number of 0 6= ki ∈ K, and among them with possibly smallest max{|βi|}where αi, βi are paths in F (E) with r(αi) = r(βi) for each i, and all βi have positivelength. It is obvious that every vertex v which is not a source of any βi for all indicesi, satisfies either rv = 0 or rv ∈ KE. In particular, one has rv, rµ ∈ KE for each sinkv in E and every path µ = µv ∈ F (E)v ending in v. Consequently, to verify condition(i) of (c) in Theorem 2.1, it is enough to see that if v is a source of any one of the pathsβi, then there are paths µj, νj ∈ F (E); r(µj) = r(νj); rmj ∈ KE such that v =

∑

µjν∗j

holds. Then v is a regular vertex, whence one can write v =∑

µiν∗i in view of Cuntz-

Krieger condition (CK2) for appropriate µi, νi ∈ F (E) satisfying v = s(µi) = s(νi) andr(µi) = r(νi). If rµi /∈ KE, then ui = r(µi) = r(νi) is not a sink by the previousremark. Therefore ui is a regular vertex and one can substitute µiν

∗i = µiuiν

∗i =

µi

∑

s(aj)=ui

aja∗jν

∗i =

∑

s(aj)=ui

(µiaj)(a∗jν

∗i ) = µiν

∗i for µiν∗i in the sum representation of

v and continue our process. After finitely many steps one obtains that every path µi

either ends in a sink, whence rµi ∈ KE, or has a length so big such that rµi is alsoa linear combination of real paths, i.e., rµi ∈ KE. This completes the verification ofcondition (i) of assertion (c) in Theorem 3.1, whence the proof is complete. �

Remarks 4.1. (1) Note the important fact that lengths of paths ending at thesame sink can go to infinity; that is, they can be arbitrarily long. On the otherhand, there could exist sinks to which any path from v does not contain a closedsubpath. To demonstrate the effect of the algorithm, writing v =

∑

µiν∗i as a

linear combination with rµi ∈ KE one can assign to v a set Evex of sinks u such

that every path from v to u does not contain closed paths, and let N(v) be themaximum length of all paths from v to sinks in Ev

ex. Let N(v) = 0 when Evex is

the empty set. These invariants may be of use for further study of Leavitt pathalgebras.

(2) For the visualization of the argument presented in the proof of the last theoremone can consider the case of the digraph


v2 a1←−v1 a2−→

a4

a3

v3 a5−→v4

and r = a2a∗3a

∗4 + a∗3a

∗2 and write down the associated expression for v1, v2.

Note the equality Ev1ex = {v2} and there are paths of arbitrary length from v1 to

the sink v4.

(3) If E is the Dynkin graphv0•

a1−→v1• −→ · · ·

vn−1

•an−→

vn• , then KE is the ring

of n × n upper triangular matrices and LK(E) is the matrix ring Kn of n × nmatrices over K. In this case LK(E) is the right maximal ring of quotients ofKE. Since KE is finite dimensional, 0 is also finite codimensional and hence thering of quotients of KE with respect to the Gabriel topology of KE is trivial.

(4) If E is the digraph •a←−

v•

b−→•, then its Leavitt path algebra is isomorphic to

K2 ⊕K2 which is again the right maximal ring of quotients of KE.(5) The proof of Theorem 4.1 also shows the importance of the Cuntz-Krieger con-

dition (CK2). One can endow LK(E) with the natural Z-graded structure. Allvertices and terms αβ∗ with α, β ∈ F (E), r(α) = r(β) and |α| = |β| are of de-gree 0. However, without (CK2), one cannot represent regular vertices as linearcombinations of proper terms αβ. Therefore, the inclusion of the quiver algebrainto the associated Leavitt algebra is neither forced to be an epimorphism norflat even for an infinite digraph with a finite set of vertices.

As the first important consequence we present a description of Leavitt path algebrassimilar to both Cuntz’ [10] construction of On(n > 1) and Raeburn’s [24] definitionof graph operator algebras which captures the naturality of Cuntz-Krieger conditions(CK1) and (CK2). One also sees the utility of our different construction. For instance inC∗-algebras, as adjoints of operators on Hilbert spaces, a∗(a ∈ E1) are defined globally,while in our algebraic setting, a∗(a ∈ E1) are defined partially, whence one needs toidentify them for a ring structure. Moreover, at the same time one can ask for anintrinsic determination of open right ideals of the quiver algebra KE given by assertion(b) of Theorem 2.1. Fortunately, it turns out that the answers for both tasks are thesame. We have seen from examples that the associated Gabriel topology admits a basisconsisting of certain finitely generated essential right ideals containing all sinks. It isfortunate that we don’t need to invoke the Gabriel topology to compute the Leavitt pathalgebra as the ring of (right) quotients of the ordinary quiver algebra. All we need is theeasy observation that the right ideal I of KE generated by all arrows together with thesinks, is an open right ideal in the Gabriel topology given in (b) of Theorem 2.1 whenceI is a dense right ideal of KE. In particular, I is even a two-sided ideal of KE. Notethat all these results are already checked as a consequence of Theorem 2.1 and Theorem4.1, but one can see it directly and easily even in the case of an arbitrary digraph.Consequently, KE can be embedded in HomKE(I,KE) by multiplication on the left


with elements of KE because the left annihilator of J is trivial, i.e., 0. In addition,as a left module over KE, HomKE(I,KE) is generated by KE and homomorphisms(functions) a∗ : J → KE for a ∈ E1 induced by sending each path γ ∈ F (E) ∩ Jto either 0 if γ does not start with a or to λ ∈ F (E) if γ = aλ. It is worth notingthat all arrows are contained in I whence for an arbitrary arrow a with u = r(a) thevertex u = a∗(a) = a∗a belongs to values of the function α∗ : J → KE, but u = r(a)belongs to J only in the case when u = r(a) is a sink. Consequently, in case of regularvertex u = r(a), a∗ is not defined on u although the functions u, a∗a are equal not onlyon uI but on uKE, too. Therefore these homomorphisms a∗(a ∈ E1) trivially satisfyCuntz-Krieger condition (CK1) for all vertices and (CK2) for regular vertices whenconsidering them as functions from I to KE. If a vertex v is a sink, then condition(CK2) becomes empty. Note the fact that for an infinite emitter v there are infinitelymany homomorphisms a∗(a ∈ s−1(v)), and so the canonical projections aa∗ provide aninfinite direct sum decomposition for the vector space vJ showing that there is roomfor using topology and restriction.

Domains of definition of partial linear transformations∑

kiµiν∗i (0 6= ki ∈ K;µi, νi ∈

F (E), r(µi) = r(νi)) from subspaces of KE into KE, exactly form the set of open rightideals for our Gabriel topology in the case of finite digraphs. In this case, it is clearthat powers I l are open and domains of definition of

∑

kiµiν∗i (0 6= ki ∈ K;µi, νi ∈

F (E), r(µi) = r(νi)) contain almost all powers I l. This implies, in view of PropositionVI.6.10 [26], that for finite digraphs E, powers of the ideal I generated by all arrows andsinks form a coarsest Gabriel topology whose localization is the Leavitt path algebraLK(E). Unfortunately, for infinite digraphs, i.e., digraphs with either infinitely manyvertices or arrows, the I-adic topology defined above is no longer a Gabriel topology.However, as we already mentioned, domains of definition of partial linear transforma-tions

∑

kiµiν∗i (0 6= ki ∈ K;µi, νi ∈ F (E), r(µi) = r(νi)) contain almost all powers I l

which are all dense ideals of KE. Therefore for an infinite digraph E the Leavitt pathalgebra LK(E) is exactly the subring of the maximal right quotient ring Qr

max(KE)generated by KE and partial functions a∗ (a ∈ E1) defined above. For finite digraphs,the Gabriel I-adic topology is Hausdorff if and only if there are no sinks. In summary,for finite digraphs one can realize LK(E) as a perfect localization of KE revealing aclose connection between their module categories. For infinite digraphs with infinite setsof either vertices or arrows, the situation is still a mystery. All we can say for certainis that LK(E) is a ring of quotients of KE in Utumi’s sense [27]. We will return indetail to the case of arbitrary digraphs in Proposition 4.7 and Corollaries 4.8, 4.9. Wesummarize the discussion above in

Corollary 4.2. Let E be an arbitrary finite digraph and I the two-sided ideal of theordinary quiver algebra KE over a field E generated by arrows and sinks. Then powersIn (n ∈ N) of I define a coarsest perfect Gabriel topology of KE whose localization is


the Leavitt path algebra LK(E). The I-adic topology is clearly Hausdorff if and only ifE has no sinks.

We now directly derive a well-known result that Leavitt path algebras of finite di-graphs are hereditary.

Proposition 4.3. A Leavitt path algebra LK(E) of a finite graph E over a field E ishereditary.

Proof. It is enough to see that every right ideal of LK(E) is projective. Namely, if J is anarbitrary right ideal of LK(E), then J = (J ∩KE)LK(E) holds in view of Theorem 2.2.Since KELK(E) is flat, one has J = (J ∩KE)LK(E) = [J ∩LK(E)]⊗KE LK(E) whencethe heredity of LK(E) follows from the fact that quiver algebras of finite digraphs arehereditary. In particular, a starting point in the theory is that every right module Mover KE admits the following standard projective resolution

0→⊕

a∈E1

Ms−1(a)⊗K r−1(a)KEf−→

⊕

v∈E0

Mv ⊗K vKEg−→ 0

where f(x ⊗ r) = x ⊗ ar − xa ⊗ x for each a ∈ E1, x ∈ Ms−1a, r ∈ r−1(a)KE andg(xv ⊗ vr) = xvr for all v ∈ E0, x ∈ M, r ∈ KE. This implies that every moduleover KE has projective dimension at most 1, that is, KE is hereditary, completing theproof. �

It is worth pointing out that the standard projective resolution described above re-mains true for a unitary module over an arbitrary digraph. However, we do not knowif this result implies the heredity of a corresponding Leavitt path algebra, even forthe Leavitt path algebra LK(1,∞) of one vertex v = 1 and infinitely many loopsai (i ∈ N). Namely, if A is a subalgebra generated by ai, then (1 − a1a

∗1)LK(1,∞) 6=

{[(1 − a1a∗1)LK(1,∞)] ∩ A}LK(1,∞) by {(1 − a1a

∗1)LK(1,∞)} ∩ A =

∑

i≥2 aiA and1 − a1a

∗i /∈

∑

i≥2 aiL(1,∞), as is routine to check. Another way to verify the heredityproperty of KE for an arbitrary digraph E is provided by Schreier’s technique as ispresented by Lewin [18] and Rosenmann and Rosset [25]. In view of its simplicity andinfluential role we recall here a detailed construction.

Let E be an arbitrary digraph and A be its ordinary quiver algebra over a field K.Then the set F (E) = ∪n≥0Fn(E) of paths in E where Fn(E) (E0 = F0(E), E1 = F1(E))is a set of paths of length 0 ≤ n ∈ N.

Schreier bases for free associative algebras introduced by Rosenmann and Rosset [25]can be extended naturally to quiver algebras of digraphs as follows.

Definition 4.2. Let A = KE be an ordinary quiver algebra of a digraph E over afield K. A Schreier basis for an arbitrary right ideal R of KE is a subset B = BR ⊆F (E) that spans a right vector space V = VR that is complementary to R (that is,A = V +R, V ∩R = 0), and which is closed to taking heads. For each n ∈ N, let Vn be


a right vector space generated by elements of B having length at most n. A Schreierbasis B is called a strong Schreier basis for R if every path in Fn(E) of length n lies inVn +R.

The argument of Rosenmann and Rosset [25] (3.2) Lemma is used to show the exis-tence of Schreier bases.

Proposition 4.4. There exists a strong Schreier basis BR for any right ideal R ofA = KE.

Proof. The case R = A is trivial because BA is just the empty set. Therefore one canassume without loss of generality that R is a proper right ideal, i.e., E0 = F0(E) 6⊆ R.A Schreier basis B = BR is constructed inductively by first taking a maximal K-linearlyindependent subset, 0B, of E0 = F0(E) modulo R and letting V0 be a K-subspace of Aspanned by 0B. Therefore V0 + R = KE0 + R = KF0(E) + R holds. If V0 + R = A,let 1B = 0B. If V0 + R 6= A, then KF1(E) is not a subspace of V0 + R. Namely,KF1(E) ⊆ V0 + R = KE0 + R would imply KF2(E) ⊆ KE0F1(E) + RF1(E) ⊆KF1(E) +R ⊆ V0 + R and so for all n, KFn(E) ⊆ V0 +R holds, hence A = V0 + R, acontradiction. Consequently, if V0 +R 6= A, then let 1B

′ be a maximal subset

{va = a | v ∈0 B a ∈ E1 = F1(E) & s(a) = v}

of F1(E) such that 1B′ is linearly independent overK modulo V0+R. Put 1B = 0B ∪ 1B

′

and let V1 be a K-space spanned by 1B. Then KE0 + KE1 = KF0(E) + KF1(E) ⊆V1 + R holds. Therefore one has the equality KF0(E) + KF1(E) + R = V1 + R.Assume now that nB

′, nB and Vn (n > 0) have been already constructed such thatKF0(E) +KF1(E) + · · ·+KFn(E) +R = Vn +R.

If Vn + R = A, let n+1B = nB. If Vn + R 6= A, then KFn+1(E) is not a subset ofVn +R because, as above, one can verify easily that KFn+1(E) ⊆ Vn +R would implyVn+R = A, a contradiction. Consequently, in case Vn+R 6= A, let n+1B

′ be a maximalsubset

{ba | a ∈ E1 = F1(E), b ∈ nB′}

which is a linearly independent set over K modulo Vn + R. Then define n+1B =

nB ∪ n+1B′ and let Vn+1 be a K-space spanned by n+1B. Hence KFn+1(E) ⊆ Vn+1+R

holds. If n+1B′ = ∅, the process stops at this step and we define BR = nB. If the

process does not stop after finitely many steps define

B = BR =∞⋃

n=0

nB .

B is clearly a strong Schreier basis of R, completing the proof. �

Schreier’s technique [18] is now suitable to show that right ideals of KE are directsums of cyclic right ideals isomorphic to cyclic right ideals vKE’s generated by verticesv ∈ E0.


Let π : A = V ⊕ R→ V be the canonical projection of A along R onto the K-spaceV spanned by a Schreier basis B of R constructed above. Then for every element x ∈ Aone has x− π(x) ∈ R, whence xy− π(x)y ∈ R holds for arbitrary elements x, y ∈ A. Inparticular, the equality

π(xy) = π(π(x)y) ∀ x, y, ∈ A(1)

holds. Consequently, for every path µ ∈ B and a ∈ E1 the element µa is either containedin B whence π(µa) = µa and so µa − π(µa) = 0, or not contained in B. In that case,by the construction of B, or equivalently, by the definition of a strong Schreier basis,0 6= µa − π(µa) ∈ R holds. Therefore, for every path µ of length l ≥ 0 in B theassociated element

uµ,a = µa− π(µa) ∈ R ∀µ ∈ B & a ∈ E1(2)

is either 0 or a nonzero element of R. We are now ready to give another proof for thefact that quiver algebras are hereditary showing the beauty of Schreier-Lewin techniquesfor digraphs.

Theorem 4.5. If R is a right ideal in a quiver algebra A = KE of an arbitrary digraphE over a field K, then R is isomorphic to a direct sum of right ideals generated byvertices.

Proof. Since the case R = A or R = 0 is obvious, one can assume without loss ofgenerality that R is a proper nonzero right ideal. Let B = BR ⊆ F (E) be a strongSchreier basis of R. We show first that the nonzero elements uµ,a = µa − π(µa) (µ ∈B, a ∈ E1) generate R. For an arbitrary path β and a ∈ E1, we have by (1)

π(β)a− π(βa) = π(β)a− π(π(β)a).

Writing π(β) =t∑

j=1

kjγj for some γj ∈ BR and kj ∈ K one has by π(β)a =t∑

j=1

kjγj

π(β)a− π(βa) = π(β)a− π(π(µ)a) =t

∑

j=1

kj(γja− π(γja)) =t

∑

j=1

kjuγj ,a.(3)

The canonical projection 1 − π of A onto R via the decomposition A = V ⊕ R sendsevery path β = c1 · · · cn ∈ F (E) (ci ∈ E1) to (1− π)(β) = β − π(β) ∈ R, which belongsalso to

∑

µ∈B& a∈E1 uµ,aA in view of (3) and the following formula

(1− π)(β) = β − π(β) =n−1∑

j=0

{π(hβ(j))cj+1 − π(hβ(j + 1))}tβ(j + 1).(4)

Consequently, by the linearity of π together with β − π(β), the image x − π(x) of anarbitrary element x ∈ A is contained in

∑

µ∈B& a∈E1 uµ,aA. Hence uµ,a (µ ∈ B; a ∈ E1)generate R.


Therefore to complete the proof, it remains to show that the sum∑

µ∈B& a∈E1 uµ,aA =

R is direct and each direct summand uµ,aA is isomorphic to a right ideal r(a)A. For thelatter claim we observe that for every path µ ∈ B and vertex v ∈ E0 a product µv iseither µ ∈ B if v = r(µ) or 0 otherwise. Hence in view of (2) for every vertex v 6= r(a)one has

µav = 0 =⇒ uµ,av = −π(µa)v ∈ R ∩ V = 0 =⇒ uµ,av = 0 = π(µa)v ∀ v 6= r(a),

whence uµ,aA is isomorphic to r(a)A. For the first claim that R is the direct sum ofsubmodules uµ,aA (µ ∈ B = BR; a ∈ E1) we use the nice argument of [[25], p. 363]instead of repeating the lengthy opaque argument of Lewin [18]. Consider finitely manynonzero elements uνj ,ai defined by νj ∈ BR and ai ∈ E1 and assume a linear dependencerelation

∑

uνj ,aixij = 0 for appropriate elements xij ∈ A. Therefore these paths νjai donot belong to B. By the above argument one can assume without loss of generality thateach xij belongs to r(ai)A, respectively. We are going to show that under this extraassumption the xij ’s are all zero. Namely, if kγ (0 6= k ∈ K) is a path of longest lengthamong paths represented as linear combinations of the xij ’s starting from s(ai), say, itis a monomial of xi j, then the term kνjaiγ cannot cancel. This is an immediate resultof the following observations. Because of the maximal length of γ, νjai is not a head of

any of the paths νjai with (νj , ai) 6= (νjai). And not being an element of B = BR, νjai is

also not a head of any path in π(νjai), which would place it in B (without any exception,even including the case (j, i) = (j, i)). This contradiction finishes the proof. �

Theorem 4.1 together with Proposition 4.3 imply also a short, elementary and directproof for the description of the Grothendieck group K0(LK(E)) of finite digraph E givenin [1] Theorem 6.1.9. For another application, recall that the Cohn path algebra CK(E)

of a digraph E is the factor of the path algebraKE subject to the Cuntz-Krieger relation(CK1). Since every Cohn path algebra CK(E) of a finite digraph E can be realized asa Leavitt path algebra of another finite digraph (see [1] Definition 1.5.16 and Theorem1.5.18), one has

Corollary 4.6. A Cohn path algebra of a finite digraph can be obtained as a ring ofquotients with respect to a perfect localization and hence it is also hereditary.

Now the Jacobson algebra of one-sided inverses can be considered as a Cohn pathalgebra of the graph with one vertex and one loop. As an extension of a polynomialalgebra K[x], it is not a flat epimorphism but we can view the Jacobson algebra ofone-sided inverses as a Toeplitz algebra, i.e., the Leavitt path algebra of the followingso-called Toeplitz graph J

a

u b−→ v


Therefore the Toeplitz algebra is a perfect localization of KJ which can be realized

as the upper triangular matrix ring R =

(

K[x] K[x]0 K

)

. Consequently, R has several

remarkable properties, and from these properties one can deduce several nice propertiesfor the Toeplitz algebra by localization, offering another route to Gerritzen’s results[13].

We shall apply results of this work to module theory over Leavitt path algebras byusing the advanced theory of modules over quiver algebras in subsequent papers. Wecomplete this section with some technical but useful properties of Leavitt path algebrasof a not necessarily finite digraph. First we present a weak form of (i) in assertion (c)of Theorem 2.1.

Proposition 4.7. For every nonzero element r ∈ LK(E) where E is an arbitrarydigraph, there is an element a ∈ KE that satisfies 0 6= ra ∈ KE.

Proof. We use induction on the minimal number l when r is represented as a linear

combination r =l∑

i=1

kiαiβ∗i (0 6= ki ∈ K;αi, βi ∈ F (E), r(αi) = r(βi)). The claim holds

obviously in the case l = 1. Assume now the claim for all integers smaller than l.Without loss of generality one can assume rv = r for some appropriate vertex v ∈ E0

and there is a path βi of positive length, otherwise r ∈ KE whence the claim follows.The minimality of l shows that the paths αi with |βi| = 0 are linearly independent, i.e.,they are pairwise different. Moreover if there is an arrow a ∈ s−1(v) \ {h(βi)}, then0 6= ra ∈ KE holds. Consequently, one can assume the equality s−1(v) = {h(βi}. If|s−1| > 1, one can write

r = x+∑

cj∈s−1(v)

zjc∗j (zj ∈ LK(E)).

Since each rcj has fewer terms than r, by the induction hypothesis, one need onlyconsider the case when all rcj = 0, whence

r = rv = r(∑

cj∈s−1(v)

cjc∗j ) = 0,

a contradiction. Therefore it remains to consider the case when ws−1(v) = {c ∈ E1} isa one element set. In this case consider the element

rc =

l∑

i=1

kiαiβ∗i c =

l∑

i=1

kiαiβ∗i (0 6= ki ∈ K;αi, βi ∈ F (E), r(αi) = r(βi))

= max{|βi|} = max{|βi|} − 1.

The equality v = cc∗ = c∗c clearly implies that r1 = rc 6= 0. Therefore, repeating theabove argument for r1 (that is, use induction on max{|βi|}) one can see after finitelymany steps that the claim for r holds and our proof is complete. �


Corollary 4.8. The (not necessarily unital) Leavitt path algebra LK(E) of an arbitrarydigraph E over a field K is a ring of right quotients of the (not necessarily unital) quiveralgebra KE in Utumi’s sense.

Proof. Consider two elements 0 6= q1, q2 of LK(E). By Proposition 4.7 there is a path αin E such that 0 6= q1α ∈ KE. If q2α belongs to KE, then we are all set. If q2α /∈ KE,then the range of α is not a sink, because for any path γ ending at a sink in KE onehas qγ ∈ KE for every element q ∈ LK(E). Hence there is an arrow a1 from the rangeof α with 0 6= q1αa1 and one can continue our process. After finitely many steps, theprocess must stop, in the sense that there must be a path β in E such that 0 6= q1β andq2β ∈ KE, completing the proof. �

As a consequence, one can reobtain [1] Proposition 2.3.7 directly in the following

Corollary 4.9. The Leavitt path algebra LK(E) of an arbitrary digraph is right non-singular.

Proof. By Proposition 4.7, for every essential right ideal J of LK(E) the right idealJ∩KE ofKE is essential. Therefore LK(E) is nonsingular by way of the non-singularityof KE which is an obvious consequence of the heredity of KE as we already remarkedafter the proof of Proposition 4.3. This completes the proof. �

Since it is well-known and, in fact, easy to describe digraphs whose quiver algebrasare either noetherian or artinian, by [26] Proposition XI.3.9 we have immediately

Corollary 4.10. If E is a finite digraph, then the Leavitt path algebra LK(E) is rightnoetherian or right artinian if and only if the quiver algebra KE has the given property.Consequently, if a Leavitt path algebra LK(E) of a finite digraph E is artinian, thenLK(E) is semisimple, and as such is a right maximal ring of quotients of KE.

Having in mind the trivial fact that LK(E) is neither right noetherian nor rightartinian if E contains an infinite emitter, by using the usual techniques of representingLeavitt path algebras as direct limits of Leavitt path algebras of finite subdigraphs, wecan again deduce characterizations of digraphs whose Leavitt path algebras satisfy someform of chain condition as they are presented in [1] Section 4.2 on pages 158 – 167.

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Renyi Institute of Mathematics, Hungarian Academy of Sciences, 1364 Budapest, Pf.

127 Hungary

Email address : [email protected]

Department of Mathematics and Computer Science, Colorado College, Colorado

Springs, CO 80903.

Email address : [email protected]

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arXiv:2108.11987v1 [math.RA] 26 Aug 2021