The Leavitt path algebras of arbitrary graphs

Gonzalo Aranda

Centre de Recerca Matemtica, Bellaterra (Barcelona)

(This is joint work with G. Abrams.)

We extend the notion of the Leavitt path algebra of a graph E

to include all directed graphs. We show how various ring-theoretic

properties of these more general structures relate to the corresponding

properties of Leavitt path algebras of row-finite graphs. Specifically,

we identify those graphs for which the corresponding Leavitt path

algebra is simple; purely infinite simple; exchange; and semiprime.

In our final result, we show that all Leavitt path algebras have zero

Jacobson radical.

Skew Polynomial Rings Over 2-Primal Rings

V.K.Bhat

SMVD University.

This article concerns the study of skew polynomial rings in terms

of 2-primal rings. 2-primal rings have been studied in recent years and

the 2-primal property is begin studied for various types of rings. In [3],

Greg Marks discusses the 2-primal property of R[x, σ, δ], where R is

a local ring, σ is an automorphism of R and δ is a σ− derivation of

R . Minimal prime ideals of 2-primal rings have been descussed by Kim

and Kwak in [2]. 2-primal near rings have been discussed by Argac

and Groenewald in [1]. Recall that a ring R is 2-primal if and only if

N(R), the set of nilpotent elements of R and P (R), the prime radical

of R are same, if and only if the prime radial is a completely semiprime

ideal. An ideal I of a rings R is called Completely semiprime if a2 ∈ I

implies a ∈ I, where a ∈ R. We also note that an Artinian ring being

reduced is 2-primal and a commutative rings is also 2-primal.

1

In this paper we show that is R is 2-primal Noetherian ring,

then R[x, σ] is also 2-primal Noetherian. Before proving the main

result, we find a relation between the minimal prime ideals of R and

those of R[x, σ], where R is a Noetherian Q-algebra and σ is an

automorphism of R .

[1 ] N.Argac and N.J. Groenewald, A generalization of 2-primal

near rings, Questiones Math-ematicae, Vol.27 No.4 (2004) 397-

413.

[2 ] N.K.Kim and T.K.Kwak, Minimal prime ideals in 2-primal

rings. Math.Japonica 50(3) (1999), 415-420.

[3 ] G.Marks, On 2-primal Ore extensions, Comm.Algebra, Vol.29

No. 5 (2001) 2113-2123.

Injective modules, spectral categories, and applications

Alberto Facchini

Padova University, Italy

We discuss some possibilities of making the injective envelope a

functor: we can consider particular rings, or specialize the modules, or

change the morphisms, that is, change the category. We shall discuss

some related concepts, like singular torsion theory and spectral cate-

gories. Then we will present some recent results due to A. Facchini

and D. Herbera. We compute some derived functors and apply our

results to the study of modules with semilocal endomorphism rings.

Every finitely presented module over a semilocal ring has a semilocal

endomorphism ring. Every finitely generated module over a commu-

tative semilocal ring has a semilocal endomorphism ring, but this is

not true for non-commutative semilocal rings.

2

The category of firm modules is not abelian

Juan Gonzalez-Ferez

Universidad de Murcia

(This is a joint work with Leandro Marin)

Let R be a nonunital ring. A right R -module M is said to

be firm if M ⊗R R → M given by m ⊗ r 7→ mr is an isomorphism.

This category generalizes the usual category of unital modules for a

unital ring and it has been used in order to study Morita Theory for

nonunital rings. It is open problem if the category of firm modules

is an abelian category. In this talk, we prove that, in general, this

category is not abelian.

Nakayama Fuller rings

S.K.Jain

Ohio University, Athens.

Nakayama (Ann. of Math. 42 (1941)) and Fuller (Pacific Jour-

nal of Mathematics 29, 1 (1969)) showed that over an artinian serial

ring every module is a direct sum of uniserial quasi-injective modules.

In particular, each right ideal of an artinian serial ring is a finite direct

sum of quasi-injective right ideals. A ring with the property that each

right ideal is a finite direct sum of quasi-injective right ideals will be

called a right Nakayama-Fuller ring. For example, commutative self-

injective rings are Nakayama-Fuller rings. In this talk, various classes

of these rings that include local, simple, prime, right non-singular right

artinian, and right serial will be discussed. Prime right self-injective

right Nakayama-Fuller rings are shown to be simple artinian. Right

artinian right non-singular right Nakayama-Fuller rings are upper tri-

angular block matrix rings over rings which are either zero rings or

division rings. The Nakayama-Fuller ring is not left-right symmetric

nor it is Morita invariant.

3

When is a Semilocal Group Algebra Continuous?

Pramod Kanwar

Ohio University - Zanesville, Ohio

(This is a joint work with S.K.Jain and J.B.Srivastava)

A ring R is said to be right CS if every right ideal of R is essential

in a summand of R , equivalently, if every closed right ideal of R is

a summand of R . A right CS ring is said to be right continuous if

every right ideal of R which is isomorphic to a summand of R is

itself a summand of R . Both right CS rings and right continuous

rings are generalizations of right selfinjective rings. It is known that

the group algebra KG of a group G over a field K is selfinjective

if and only if G is a finite group. If a continuous group algebra KG

of a group G over a field K is continuous then G is a locally finite

group. Among others, it is shown that (i) a semilocal (semiperfect)

group algebra KG of an infinite nilpotent group G over a field K

of characteristic p > 0 is CS (equivalently continuous) if and only if

G = P × H , where P is an infinite locally finite p -group and H is

a finite abelian group whose order is not divisible by p , (ii) if K is a

field of characteristic p > 0 and G = P ×H where P is an infinite

locally finite p -group (not necessarily nilpotent) and H is a finite

group whose order is not divisible by p then KG is CS if and only

if H is abelian. Furthermore, commutative semilocal group algebras

and local PI group algebras are always continuous.

Rook version of the class partition algebras

A.Joseph Kennedy

Pondichery University, Pondichery

The class partition algebras Pk(m, n) arise from looking at what

commutes with the restricted action (Centralizer) of the wreath prod-

uct group Sm o Sn on W⊗k , where W is the natural representation

4

of Smn . The rook (half) Partition algebra RPk(x) is the centralizer

algebra of the symmetric group Sn on U⊗k , where U is the direct

sum of the natural representation and the trivial representation of Sn .

This rook version is used to construct the RSK correspondence for the

partition algebras. In this paper, we study the rook version of the class

partition algebras.

On Cancellation Properties of Modules.

Dinesh Khurana

Panjab University, Chandigarh.

We will discuss some cancellation properties of modules and re-

lationships between them.

Alternating Matrices over Rings and Fields

T. Y. Lam and Richard G. Swan

University of California, Berkeley, USA

This talk concerns the theory of alternating matrices over rings

and fields. A square matrix M is said to be alternating-clean if

M = A+U where A is alternating and U is invertible. (This notion

was motivated by the the problem of finding ”sections” in projective

modules.) Over fields and division rings, we show that, with some

rather obvious exceptions, all matrices are alternating-clean. Such

classification work does not seem possible over commutative rings.

However, alternating matrices and their Pfaffians make perfect sense

over commutative rings, and one may hope to prove theorems about

them. In this direction, we’ll report some recent results on the (Mc-

Coy) rank of alternating matrices, and on the relationship between

the McCoy rank and the ”principal rank”. If time permits, we’ll also

say something about von Neumann regular matrices over commutative

rings.

5

Wedderburn polynomials and their applications

Andre Leroy

University of Artois, France

We will first introduce Wedderburn polynomials and some of

their characterizations in the context of an Ore extension over a di-

vision ring. We will then present some applicatios such as : diago-

nalizations and triangulations of matrices over division rings, left and

right roots of Ore polynomials, relations with noncommutative sym-

metric functions, factorizations of differential equations. We will end

the talk considering some generalizations such as the fully reducible

polynomials or working with more general rings than Ore extensions.

Right Continuous Functors over Firm Modules

Leandro Marin

Universidad de Murcia

Let R be a nonunital associative ring. A left R -module M is

said to be firm if R ⊗R M → M is an isomorphism. In this talk

we consider the right continuous fuctors between the category of firm

modules and abelian groups. This structure provides consequences in

Morita Theory for nonunital rings.

Investigating OD-Characterizability of some finite groups

Ali reza Moghaddamfar

K.N. Toosi University of Technology, Iran

We already introduced the notion of degree pattern of a finite

group G and it is proved that the following simple groups are uniquely

determined by their degree patterns and orders: all sporadic simple

groups, alternating groups Ap where p ≥ 5 is a twin prime and some

6

simple groups of Lie type. In this lecture, we will continue this inves-

tigation. In particular, we will show that the automorphism groups

of sporadic simple groups (except Aut(J2) and Aut(M cL) ), all C22 -

simple groups, the alternating groups Ap , Ap+1 , Ap+2 and the sym-

metric groups Sp , Sp+1 where p is a prime number, are also uniquely

determined by their degree patterns and orders.

On Goldie prime ideals of skew polynomial rings

A. Moussavi

Tarbiat Modarres University,Tehran-Iran

(This is a joint work with A. Ahmadi)

Let R be a ring with an injective endomorphism α of R . Then

every prime ideal of the skew Laurent polynomial ring R[x, x−1; α] is

left Goldie if and only if every strongly α -prime ideal of the Jordan

extension A(R,α) of R is a finite intersection of left Goldie prime

ideals. If R is a Noetherian ring, then all prime ideals of the skew

polynomial ring R[x; α] is left Goldie.

References

[1] S. A. Amitsur, Radicals of polynomial rings, Canad. J. Math.8

(1956) 355-361.

[2] S. S. Bedi and J. Ram, Jacobson radical of skew polynomial rings

and group rings, Israel J. Math. 35 (1980) 327-338.

[3] A. D. Bell, When are all prime ideals in an Skew polynomial ring

Goldie?, Comm. Algebra, 13 (8) (1985) 1743-1762.

[4] G. Cauchon and J. C. Robson, Endomorphisms, derivations and

polynomial rings, J. Algebra 53 (1978) 227-238.

[5] A. W. Goldie and G. Michler, Skew polynomial ring and polycyclic

group rings, J. London Math. Soc. 9 (2) (1974) 337-345.

[6] K. R. Gooderl, R. B. Warfield “An introduction to noncommutative

7

Noetherian rings, Cambridge University Press, Cambridge, (1989).

[7] A. Jategaonkar, Skew polynomial rings over orders in Artinian

rings, J. Algebra 21 (1972) 51-59.

[8] D. A. Jordan, Bijective extensions of injective ring endomorphisms,

J. London Math. Soc. (2) 25 (1982), 435-448.

[9] A. Moussavi, On the semiprimitivity of skew polynomial rings.

Proc. Edinburgh Math. Soc. 36 (2) (1993) 169-178.

Invariant Ideals of Abelian Group Algebras under the

Multiplicative Action of a Division Ring and the Final

Value Problem

J.M.Osterburg

UC, Cincinnati, Ohio, USA

Let D be a division ring and let V = Dn be a finite dimen-

sional right D -vector space. If G = D• is the multiplicative group

of D, then G acts on V via scalar multiplication. Hence G acts

on the group algebra of V over K. If char(K) 6= char(D), then we

completely describe the G -stable ideals of V. These results follow

from corresponding work of C. J. B. Brookes and D. M. Evans for the

rational numbers and for infinite locally finite fields by work of D. S.

Passman and A. E. Zalesski. The result for division rings is a going-

up and going-down type of theorem due to the author, Passman and

Zalesski.

We will then turn to polynomial forms, which are not necessarily

linear maps, from infinite modules over the ring Z to a finite abelian

group that satisfy a homogeneous property and a derivative property.

Let f be such a form. Amongst all submodules B of finite index,

there is one, B, with |f(B)| of minimal size and we call f(B) the

final value of f. Passman asked if the final value had to be a subgroup.

8

The answer to this question is yes, if the form has degree ≤ 2 or if

the infinite module is a fg abelian group, by my work, but in general,

the answer is no. We finally present our example of a polynomial form

of arbitrary degree that is nonzero on every submodule of finite index

and, then we give the surprisingly easy example (due to Passman) of

a polynomial form which has a final value that is not a subgroup.

Group Homology and Higher Traces

Inder Bir S.Passi

Panjab University, Chandigarh

Let k be a commutative ring with identity and A a k− algebra.

If M is a k− module, then a trace map τ : A → M is a k− linear

map satisfying τ(ab) = τ(ba) for a, b ∈ A. Let 0 → I → R → 0 be

an algebra extension. The trace maps on R/In are called the higher

traces on A relative to the algebra extension R/I ' A. The universal

n the higher trace map is the natural projection τn : R/In → R/(In +

[R,R]), where [R,R] is the additive subgroup of R spanned by the

elements rs − sr(r, s ∈ R). D.Quillen (1989) has given a description

of the cyclic homology groups HC?(A) of A in terms of the inverse

limits, over the category of algebra extensions, of the k− modules

R/(In+1 + [R,R]) and In+1/[In, I] :

HC2n ' lim←

R/(In+1 + [R,R]), HC2n+1(A) ' lim←

In+1/[In, I].

I will discuss recent work with I . Emmanouil (Athens) and

with R.Mikhailov(Moscow) on group homology that is motivated by

this description of cyclic homology and the subsequent work of Cuntz-

Quillen (1995).

9

Linear Groups and Group Rings

Donald S. Passman

University of Wisconsin, Madison, USA

This is joint work with Jairo Goncalves, and concerns the exis-

tence of free subgroups in unit groups of matrix rings and group rings.

The subject starts with the famous ping-pong lemma of F. Klein and

perhaps reaches its high point with the theorem of J. Tits on free

subgroups of linear groups. There are interesting group ring results

due to B. Hartley and P. F. Pickel, as well as to Z. S. Marciniak and

S. K. Sehgal. Indeed, much of Sehgal’s group ring books are concerned

with this problem. Our work generalizes aspects of Tits’ machinery

and then applies it to obtain concrete pairs of units that generate free

groups in the group rings of certain critical groups. The hard part

here is to verify the so-called idempotent condition.

The Cuntz semigroup: representations and applications to

the classification program and the Blackadar-Handelman

conjectures.

Francesc Perera

Universitat Autnoma de Barcelona

(This is a joint work with Nate Brown and Andrew Toms)

For a large class of unital and exact C*-algebras, we identify

the Cuntz semigroup in terms of the projection monoid and a certain

semigroup of functions defined on the space of traces. This resolves

two conjectures of Blackadar and Handelman and offers significant

conceptual insight into Elliott’s classification program.

10

Group Algebras satisfying a certain Lie Identity

Meena Sahai

Lucknow University, Lucknow

Let K be a field and let G be a group. We denote by L(KG)

the associated Lie algebra of the group algebra KG under the Lie

multiplication [x, y] = xy − yx ; x, y ∈ KG . The group algebra KG

is Lie metabelian if the associated Lie algebra KG is metabelian. Also

KG is Lie centrally metabelian if L(KG) is centrally metabelian. Lie

metabelian group algebras and Lie centrally metabelian group algebras

have already been studied by various authors. Continuing in this di-

rection further we have characterized group algebras KG satisfying

the Lie identity [[x, y].[u, v], [z, t]] = 0 for all x, y, u, v, z, t ∈ KG in

case Char K 6= 2 .

The main result of this paper is that when G is a group and

K is a field of characteristic 6= 2 then KG satisfies the above Lie

identity if and only if one of the following holds: (i) G is abelian; (ii)

Char K = 5 , G′ is central cyclic of order 5; (iii) Char K = 3 and

G′ is cyclic of order 3 or G is nilpotent of class 2 and G′ = C3 ×C3 .

G -Prime ideals in semirings and their skew group semirings

Ram Prakash Sharma

Himachal Pradesh University,Shimla

(This is a joint work with Tilak Raj Sharma)

The study of groups acting on rings was initiated as an attempt

to develop Galois theory for noncommutative rings. The theory was

initially extended to division rings by N. Jacobson in 1940. The con-

ditions in case of general noncommutative rings became complex and

so a fresh approach was made by starting with some simple questions

regarding the relationships of the structure of a ring R with identity

11

to the structure of the fixed subring RG with respect to the finite

automorphism group G . A useful tool in this subject is the skew

group ring R ∗ G of all formal sums∑g

rgg, rg ∈ R, g ∈ G . In fact

R ∗G is an associative ring containing R , G and RG , so the results

of skew group rings are therefore surely of interest. The researchers in

this direction have settled many questions regarding the relationships

of the structure of a ring R to the structure of the fixed subring RG

and to the skew group ring R ∗G . The Going down problem is one of

the main questions settled by M. Lorentz and D.S. Passman in their

paper in1979 regarding the relationships of G -prime ideals of a ring

R to the prime ideals of R ∗ G. We start this paper with the aim to

achieve its analogue in a semiring with finite group action on it.

In the absence of additive inverses, we need a weaker condition,

i.e. cancellation of the elements, so throughout this paper we assume

that semiring R is additively cancellative. If R is an additively can-

cellative semiring, then R is isomorphic to a subsemiring of a ring R∆

such that every element of R∆ is the difference between two elements

in the image of R . The action of G can be extended to R∆ and so it

becomes a useful tool to study the ideals of R and R∗G . It is evident

from (Golan, 1999) that there are plenty of such semirings. We also

have to impose another weak version of the condition of having ad-

ditive inverses, that is, R is assumed to be yoked semiring whenever

it is required. The semirings N (the set of nonnegative integers) and

Q+ (the set of nonnegative rational numbers) are surely additively

cancellative and yoked.

The main result derived herein is as follows:

If R is an additively cancellative yoked semiring and G a finite

group acting on R , then(i) If A is a subtractive G -prime ideal of R ,

then there exists a prime ideal P of R ∗G such that P ∩R = A .

12

(ii) If A1, A2 are subtractive G -prime ideals of R with A1 ( A2

, then there exist two prime ideals P1 and P2 in R ∗ G such that

P1 ∩R = A1 , P2 ∩R = A2 with P1 ( P2 .

The maximal algebra of quotients of some Lie algebras

Mercedes Siles

Universidad de Malaga.

(This is a joint work with Matej Bresar, Francesc Perera and Juana

Sanchez Ortega.)

We describe maximal algebras of quotients of Lie algebras of

the form A−/Z(A) , for A an associative algebra ( Z(A) denotes the

center of A ). Other questions that arise naturally are also answered.

The first one is related to the coincidence of the maximal algebra of

quotients of an essential ideal and that of the algebra. The second

one is if the maximal algebra of quotients of the maximal algebra of

quotients of a Lie algebra is the maximal algebra of quotients.

Affine Algebras, GK Dimension and Primitivity

Lance Small

University of California, San Diego, USA

We will discuss the primitivity of certain algebras like the en-

veloping algebra of the centerless Virasoro algebra and its consequences.

Other questions relating to algebras of low dimension will also

be considered.

13

Group Algebras of infinite groups having certain trivial

torsion subgroups

J.B.Srivastava

Indian Institute of Technology, Delhi

Let KG be the group algebra of a group G over a field K . It

is well known that KG is prime if and only if ∆+(G) = (1) , where

∆+(G) is the torsion subgroup of the FC -subgroup ∆(G) of G .

Strongly prime rings have been studied by Handelman and Lawrence

(1975). They have shown that if KG is strongly prime then the locally

finite radical, L(G), is trivial. It is conjectured that KG is strongly

prime if and only if L(G) = (1) . This conjecture has been verified for

several classes of groups. We completely characterize group algebras

KG for which Λ+(G) = (1) where Λ+(G) is a torsion subgroup of G

such that ∆+(G) ⊆ Λ+(G) ⊆ L(G) . The class of ∗ -prime algebras

has been introduced. This class of algebras lies strictly between the

class of prime and strongly prime algebras. We prove that a group

algebra KG is ∗ -prime if and only if Λ+(G) = (1) . We study group

algebras of wreath products of groups and use intersection theorems

to obtain certain results showing the importance of ∗ -prime group

algebras. A brief review of corresponding radicals is also presented.

Rings generated by units.

Ashish K.Srivastava

Ohio University, Athens

(This is a joint work with D. Khurana)

A classical result of Zelinsky states that every linear transforma-

tion on a vector space V , except when V is one-dimensional over Z2 ,

is sum of two invertible linear transformations. We extend this result

to any right self-injective ring R by proving that every element of R

14

is sum of two units if and only if no factor ring of R is isomorphic to

Z2 . We also give a complete characterization of unit sum numbers of

right self-injective rings.

From Temperley-Lieb algebras to non-crossing partitions

V.S. Sunder

Institute of Mathematical Sciences, Chennai, India

After introducing Temperley-Lieb algebras, we commence by

identifying the dimensions of these algebras with the Catalan num-

bers, as well as with the number of non-crossing partitions. We then

demonstrate that the algebras TL2n(δ) and NC2n(δ2) are isomorphic

(by a non-obvious isomorphism), this proof relying on a ’linearisation

result’

Cayley-Hamilton theorem for matrices over an arbitrary

ring

Jeno Szigeti

Institute of Mathematics, University of Miskolc, 3515 Hungary

The characteristic polynomial

p(x) = λ0 + λ1x + ... + λn−1xn−1 + n!xn

of an n × n matrix A ∈ Mn(R) can be defined in R[x] by using a

canonical construction (here R is an arbitrary unitary ring). Then

we obtain a Cayley-Hamilton identity with right matrix coefficients of

the following form:

(λ0I +C0)+A(λ1I +C1)+ ...+An−1(λn−1I +Cn−1)+An(n!I +Cn) = 0,

where I ∈ Mn(R) is the identity matrix and the entries of the n× n

matrices Ci , 0 ≤ i ≤ n are in the additive subgroup [R,R] of R

15

generated by the commutators [x, y] = xy − yx with x, y ∈ R (a

more precise description of the entries in the Ci ’s can be deduced

from the proof). A similar identity with left matrix coefficients can

be obtained analogously. If R is commutative, then C0 = C1 = ... =

Cn−1 = Cn = 0 and our identity gives the n! times scalar multiple of

the classical Cayley-Hamilton identity for A .

Mono-injective and epi-projective modules

N.Vanaja

146, Row House, Sevija, Sector 3, Charkop, Kandivli (W), Mumbai

A module M is called mono-injective if for every N ⊆ M any

monomorphism f : N → M can be extended to M . We characterise

mono-injective extending modules. Dually we call M epi-projective

if for every N ⊆ M any epimorphism f : M → M/N can be lifted

to M . A module M is called direct projective if, for every direct

summand X of M , every epimorphism M → X splits. A direct

projective lifting module is called a discrete module. We characterise

discrete and epi-projective lifting modules in terms of lifting of maps

and prove some of their properties.

Clean, almost clean, potent commutative rings

K.Varadarajan

University of Calgary, Canada

K.Samei obtained some resuts tying up cleanness of a commuta-

tive ring R with zero-dimensionality of Max(R) in the Zariski topol-

ogy ( Comm. Alg. 32(9), 2004, pp 3479-3486 ). In some recent work of

mine these results have been considerably strengthened. In the class of

rings C(X) , with X a Tychonoff space W.Wm Mc.Govern ( Comm.

Alg. 31(7), 2003, pp 3295-3304 ) has charcterised potent rings as the

16

ones with X admitting a clopen π -base. We prove the analogous

result for any commutative ring in terms of the Zariski topology on

Max(R) . Mc.Govern also introduced the concept of an almost clean

ring and proved that C(X) is almost clean if and only if it is clean.

We prove a similar result for all commutative Gelfand rings R with

J(R) = 0 . In my talk I will present these results.

On the structure of the nonsemisimple Brauer algebra

Hans Wenzl

University of California, San Diego, USA

We give a new proof of the restriction rules for representations of

Gl(N) to O(N) . The result is expressed in terms of a reflection group,

suggested by a similar result for tilting modules of quantum groups.

We obtain from this a precise conjecture about the structure of the

nonsemisimple Brauer algebra in terms of parabolic Kazhdan-Lusztig

polynomials.

Hopf monads for categories

Robert Wisbauer

Heinrich Heine University, Dusseldorf, Germany

Let F and G be endofunctors of categories A and B , respec-

tively. Denote by AF and BG the categories of F - and G -algebras

with forgetful functors UF : AF → A and UG : BG → B .

Given functors T : A → B and T : AF → BG , one says that T

is a lifting of T provided they yield a commutative diagram

AFT //

UF

��

BG

UG

��A T // B.

17

Distributive laws are related to lifting properties of this type. As

an example one may consider the Hopf monads as defined by Moerdijk

(in [3]).

In this language, entwining natural transformations (mixed dis-

tributive laws) between monads and comonads on arbitrary categories

were first considered by van Osdol in [4]. As a special case the entwin-

ing of algebras and coalgebras over commutative rings are described

which yields examples of corings (see [2]).

It is outlined that these techniques allow to generalize the notion

of bialgebras and Hopf algebras from module categories to arbitrary

categories.

References.

1. Beck, J., Distributive laws, Seminar on Triples and Categorical Homol-ogy Theory, B. Eckmann (ed), Springer LNM 80, 119-140 (1969)

2. Brzezinski, T. and Wisbauer, R., Corings and Comodules, London Math.Soc. Lect. Note Ser. 309, Cambridge Univ. Press, Cambridge (2003)

3. Moerdijk, I., Monads on tensor categories, J. Pure Appl. Algebra 168(2-3), 189-208 (2002)

4. van Osdol, D. H., Sheaves in regular categories, in: Exact categoriesand categories of sheaves, Springer Lecture Notes Math. 236, 223-239(1971)

Cellular algebras and twisted semigroup algebras

Changchang Xi

Beijing Normal University

Cellular algebras are defined by the existence of a basis with

certain multiplicative properties by Graham and Lehrer. It turns out

that a large variety of associative algebras falls into this class. Typical

18

examples include many diagram algebras such as Brauer algebras and

partition algebras, and group algebras of symmetric groups, Hecke

algebras of type A and q -rock monoids. One of the advantages of

cellular algebras over a field is that the representation theory can be

studied by linear algebra. In this talk, we shall first report some basic

results on cellular algebras, and then show that the cellular structure

appears also in (twisted) semigroup algebras. Here I shall mainly

report some recent results of myself jointly done with X.J.Guo, and of

others.

[1 ] X. J. Guo and C. C. Xi, Cellularity of twisted semigroup

algebras. Preprint, 2006.

[2 ] J. East, Cellular algebras and inverse semigroups.J. Algebra

296(2006), 505-519.

[3 ] S. Wilcox, Cellularity of diagram algebras and twisted semi-

group algebras. Preprint, 2005.

[4 ]S. Konig and C.C. Xi, A characteristic-free approach to

Brauer algebras. Trans. Amer. Math. Soc. 353(2001), 1489-

1505.

[5 ]J.J. Graham and G.I. Lehrer, Cellular algebras. Invent.

Math. 123(1996), 1-34.

Net subrings of generalized matrix rings

A.V.Yakovlev

Universitetsky prospekt, St.Petersburg

Let Λ be a ring with the following properties: a) Λ is a di-

rect sum of left ideals P1, . . . , Pn ; b) every non-trivial homomorphism

Pi → Pj is a monomorphism; c) for every i, j the intersection of any

19

two submodules of Pj isomorphic to Pi contains a submodule isomor-

phic to Pi . We prove that then Λ can be represented as a subring

associated with a net of ideals in a generalized matrix ring.

The class of rings which we consider here contains, for exam-

ple, semiherditary semiserial rings, i.e., the rings that are both direct

sums of left ideals with linear lattices of subideals and direct sums

of right ideals with linear lattices of subideals. But it also contains

one-sided semiserial semiherditary rings, and is, in fact, much larger

than these well-investigated classes of rings. Many results concerning

semiherditary semiserial rings are particular cases of our theorem.

A Note on Admissible groups

Alireza Zokayi

K.N. Toosi University of Technology

Tehran, IRAN

A complete map for a group G is a permutation ϕ : G −→ G

such that g 7−→ gϕ(g) is a permutation of G . A group G possessing a

complete map is said to be admissible. A conjecture of M. Hall and L.

J. Paieg states that every group of even order with non-cyclic Sylow

2-subgroup is always admissible. In this lecture, first we show that

every non-solvable group with dihedral Sylow 2-subgroup is admissible.

Next, we will give a different approach to the admissibility of Mthieu

groups M23 and M24 using their exact factorizations. Later on, we

will prove that if a non-simple group G has a factorization G =

AB � A × B with A, B simple groups such that A is maximal and

admissible then G itself is admissible.

20

Paper Presentation

On Generalized Jordan derivations in prime rings

Mohammad Ashraf

Aligarh Muslim University, India

Let R and S be associative rings and θ, φ be homomorphisms of S

into R . An additive mapping d : R → R is aid to be a derivation on

R if d(ab) = d(a)b + ad(b) holds for all a, b ∈ R . Suppose that M

is an R -bimodule. An additive mapping d : S → M is said to be a

(θ, φ) -derivation (resp. Jordan (θ, φ) -derivation) if d(ab) = d(a)θ(b)+

φ(a)d(b) (resp. d(a2) = d(a)θ(b) + φ(a)d(b) ) holds for all a, b ∈ S .

Let M be 2 -torsion free such that mRx = {0} with m ∈ M, x ∈ R

implies that either m = 0 or x = 0 . An additive mapping F :

S → M is called a generalized (θ, φ) -derivation (resp. Jordan (θ, φ) -

derivation) if there exists a (θ, φ) -derivation d : S → M such that

F (ab) = F (a)θ(b) + φ(a)d(b) (resp. F (a2) = F (a)θ(b) + φ(a)d(b) ),

holds for all a, b ∈ S . In the present paper, it is shown that if θ is one-

one and onto, then every generalized Jordan (θ, φ) -derivation F is a

generalized (θ, φ) -derivation. Further some more related results are

also obtained. In fact our results unify and generalize several results.

Structure of certain rings

Asma Ali

Aligarh Muslim University, Aligarh (INDIA)

Using commutativity of rings satisfying (xy)n(x,y) = xy proved

by Searcoid and MacHale [Amer. Math. Monthly 93(1986), 121 - 122],

Ligh and Luh [Amer. Math. Monthly 95(1989), 223 - 228] have given

a direct sum decomposition for rings with the mentioned condition.

Further Bell and Ligh [Math. J. Okayama Univ. 31(1990), 93 - 99]

21

sharpened the result and obtained a decomposition theorem for rings

with the property xy = (xy)2f(x, y) where f(X,Y ) ∈ Z < X, Y > ,

the ring of polynomials in two noncommuting indeterminates.In the

present paper we continue the study and investigate structure of cer-

tain rings satisfying the following condition which is more general than

the mentioned conditions : xy = p(x, y) , where p(x, y) is an admis-

sible polynomial in Z < X, Y > . Moreover we deduce the commuta-

tivity of such rings.

Subrings of FGI-Rings

Mamadou Barry

Universit Cheikh Anta Diop, Dakar, Sngal

(This is a joint work with Mamadou Sanghare, Sidy Demba Toure)

Let R be a noncommutative associative ring with unity 1 6= 0 .

A left R-module RM is said to have property (I) (resp.(I)), if every

injective (resp.surjective ) endomorphism of RM is an automorphism

of M .It is well known that every Artinian (resp. Noetherian )mod-

ule satisfies property (I)(resp.(S))and the converse is not true.A ring

R is called left I-ring (resp.S-ring) if every left R -module with prop-

erty (I)(resp. (S)) is Artinian (resp.Notherian ). A ring R is called

left (right) FGI-ring if every left (right) R-module with property (I) is

finitely generated. R is called FGI-ring if it is both a left and right FGI-

ring. If R is either commutative or a duo ring then the class of S-rings,

I-rings ,FGS-rings and FGI-rings are exactly the class of Artinian prin-

cipal ideal rings(see [2],[3] and [4]). let R be an integral domain and K

be its classical quotient field .If R 6= K , then K is a FGI-ring but R is

not a FGI-ring . It is known that a subring B of a left FGI-ring is not

in general a left FGI-ring even if R is a finitely generated B -module,

for example the ring M3(K) of 3 × 3 matrices over a field K is a

22

left FGI-ring whereas its subring B =

( α 0 0

β α 0

γ 0 α

)/α, β, γ ∈ K

which is a commutative ring with a non principal Jacobson radical

J(B) = K.

0 0 0

1 0 0

0 0 0

+ K.

0 0 0

0 0 0

1 0 0

is not a FGI-ring (see

[5],theorem.8). A ring is said to be a ring with polynomial identity

(P.I-ring )if there exists a polynomial f(X1, X2, ..., Xn), n ≥ 2 , in the

non commuting indeterminates X1, X2, ..., Xn over the center Z of

R such that one of the monomials of f of the highest total degree

has coefficient 1, and f(a1, a2, ..., an) = 0 for all a1, a2, ..., an in R .

Throughout this paper all rings considered are associative rings with

unity , and by a module M over a ring R we always understand

an unitary left R -module. We use MR to emphasize that M is an

unitary right R -module. The main result of this note is the follow-

ing theorem: Let R be a left Artinian FGI-ring and B be a subring

of R contained in the center Z of R . Suppose that R is a finitely

generated flat B -module. Then B is a FGI-ring.

Algebras with skew-symmetric identity of degree 3

A.S. Dzhumadil’daev

Institute of Mathematics, Almaty

Algebras with one of the following identities are considered:

[[t1, t2], t3] + [[t2, t3], t1] + [[t3, t1], t2] = 0, (Lie-Admissible)

[t1, t2]t3 +[t2, t3]t1 +[t3, t1]t2 = 0, ( 0 -Lie-Admissible (shortly 0 -Alia))

{[t1, t2], t3}+{[t2, t3], t1}+{[t3, t1], t2} = 0, ( 1 -Lie-admissible (shortly 1 -Alia))

23

where [t1, t2] = t1t2 − t2t1 and {t1, t2} = t1t2 + t2t1. We prove that

any algebra with a skew-symmetric identity of degree 3 is isomorphic

or anti-isomorphic to one of such algebras or can be obtained as their

q -commutator algebras.

Example 1. (C[x], ◦) under multiplication a ◦ b = ∂(a)∂2(b) is 1 -

Alia and simple.

Example 2. (C[x], ?), where a?b = ∂3(a)b+4∂2(a)∂(b)+5∂(a)∂2(b)+

2a ∂3(b), is 0 -Alia and simple.

Cocyclically Copure Submodules

Seema S.Gramopadhye

Karnatak University, Dharwad, India

(This is a joint work with V.A. Hiremath)

P.M.Cohn[3] introduced the notion of pure submodules. Dual to

the notion of purity, the first author [6] defined the copurity, for the

category of modules using cofinitely related modules. Later, James

Simmons [13] studied the cyclic purity as the generalization of purity.

In this paper we study the cocyclic copurity as the dual to the cyclic

purity. Also we have studied cocyclic copurity in relation to solvability

of equations and also intersection purity is studied.

Around the structure of an M-HNP module M

Irawati

Institut Teknologi Bandung, Bandung, Indonesia

The notion of an M-HNP module is introduced as a generaliza-

tion of the HNP ring. It is obvious that an HNP ring is an R-HNP

module. We present several results related to M as an M-HNP mod-

ule, including the structure of End(M), the structure of M/N with N

24

as an essential submodule of M, and the decomposition of a finitely

M-generated module in sigma[M].

Relative Character Graph: A variation on a theme of

Brauer

A.Vincent Jeyakumar

Ponnaiyah Ramajayam College, Thanjavur.

(This is a joint work with R.Stella Maragatham)

This paper is the outcome of the authors attempt to construct

some new finite simple graphs using complex irreducible characters of

finite group G and to study some of the properties of these graphs.

Our title may be justified in some sense, because, the vertices of this

graph Γ(G, H) (which we call the Relative Character Graph relative

to a subgroup H of G abbreviated as RC graph) are the same as

those of the celebrated Brauer graphs, which were studied by Richard

Brauer in the early 1940’s. The adjacency conditions in the Brauer

graph construction and our construction of Γ(G, H) differ of course.

In the Brauer graph case, a prime p dividing O(G) is involved, and in

our RC -graph case, a subgroup H is involved.(The beauty is, when

the subgroup H is a p− subgroup with certain additional conditions,

both these graphs coincide!).

Some basic properties like connectivity, triangulation etc of Γ(G, H)

are already obtained by some students of the first author. In this pa-

per we give special attention to the complement of Γ(G, H) . It is

very rare that the complement is also an RC− graph (relative to some

subgroup K ). We discuss a special case when the complement is also

an RC− graph. We naturally raise an open question regarding char-

acterization of subgroups H (of an arbitrary group G ) wherein both

Γ(G, H) and its complement are connected. However, we prove that,

when q and n denote the number of edges and vertices of Γ(G, H) re-

25

spectively, when the graph is not a tree, when the right action of G on

G/H is doubly transitive and if q ≤ (n− 1)C2 , then the complement

is connected.

We also discuss briefly yet another question as to some suitable

conditions on an arbitrary finite simple graph which arises as an RC -

graph with respect to some finite group G and subgroup H . We

call such graphs as RC− traceable. We conclude the paper by giving

several typical examples.

These RC− graphs perhaps open the gateway to further study

of group characters via graph theory. Such study can include signed

graphs, Domination theory of RC− graphs and perhaps an attempt

closer towards a solution of the famous ’Vizing’s Conjecture’ involving

product graphs.

∗ -prime radical of group algebras

Kanchan Joshi

Indian Institute of Technology, Delhi

A new class of algebras called ∗ -prime algebras have been intro-

duced in our recent paper. An associative algebra R is called ∗ -prime

if for every 0 6= r ∈ R there exists a finitely generated subalgebra S

of R such that rSt = 0 implies t = 0 . This class lies strictly between

the class of prime algebras and strongly prime algebras. We have also

defined the ∗ - prime radical of an algebra R as the intersection of all

∗ -prime ideals of R . We give the elementwise characterization of of

the ∗ -prime radical by defining the ∗ -p system. Also the relationship

of this radical with other well known radicals is discussed. We also

study the structure of the ∗ -prime radical and strongly prime radical

of the group algebra KG of the group G over the field K .

26

Prime ideals of a ring R with Brauer diagrams acting on it

Vikram Singh Kapil

Himachal Pradesh University, Shimla, India

(This is a joint work with Ram Parkash Sharma)

The study of groups acting on rings was initiated as an attempt

to develop Galois theory for noncommutative rings. The theory was

initially extended to division rings by N.Jacobson in 1940. The con-

ditions in case of general noncommutative rings became complex and

so a fresh approach was made by starting with some simple questions

regarding the relationships of the structure of a ring R with identity

to the structure of the fixed subring RG with respect to the finite au-

tomorphism group G . A useful tool in this subject is the skew group

ring R ∗ G . The researchers have settled many questions regarding

the relationships of the structure of a ring R to the structure of the

skew group ring R ∗G . The Orbit problem, Incomparability and the

Going down problem are the main questions settled by M.Lorentz and

D.S.Passman in 1979 regarding the relationships of G-prime ideals of

a ring R to the prime ideals of R ∗G .

Let Dn be the set of diagrams, which consist of two rows of

n points labelled {1, 2, · · · , n} , with each dot joined to precisely one

other dot (distinct from itself). Dn becomes monoid under the prod-

uct of two diagrams d1 and d2 by putting d1 above d2 and joining

corresponding points ignoring the interior loops. Let R be a ring

with identity 1 such that Dn acts on it, i.e. (rdi)dj = rdidj and

re = r, r ∈ R, di, dj ∈ Dn . Further we assume that r 7→ rd is a ring

homomorphism from Rdi, dj 7→ Dn

For each diagram d ∈ Dn , there is a unique diagram d∗ ∈ Dn ,

the reflection of d in a horizontal axis, satisfying :

27

(∑

rdidi)(

∑rdjdj) =

∑(∑

rdir

d∗idj

)dk.

Moreover, R ∗Dn is a free R-module with basis di|di ∈ Dn.

The set Sn of all Brauer diagrams without horizontal edges can

be regarded as the permutation group on {1, 2, · · · , n} and acts on R

as a subgroup of its automorphisms. The skew group ring R ∗ Sn is a

subring of R ∗Dn . Thus the known results for R with finite group Sn

acting on it are applied to settle the Incomparability and the Going

down problem for R and R ∗Dn.

Remarks on generalized derivations of semi prime rings

Moharram A. Khan

Eritrea Institute of Technology, Asmara, ERITREA

In this talk, we first introduce a generalized derivation on a prime

ring which acts as a homomorphism or an anti-homomorphism on the

non-zero one sided ideal in the ring is vanish. Secondly, we discuss

a correspondence between generalized derivations related to a fixed

derivation and left multipliers. Finally, we provide a characterization

of generalized derivations on semi prime ring.

28

Prime Antiflexible Derivation Alternator Rings

V.Maheswara Rao

Sri Krishnadevaraya University, Anantapur-515003,A.P.

(This is a joint work with K.Suvarna)

A non-associative ring with characteristic not 2 ia called a derivation

alternator ring if it satisfies the following identities:

(x, x, x) = 0,

(yz, x, x) = y(z, x, x) + (y, x, x)z,

and(x, x, yz) = y(x, x, z) + (x, x, y)z,

where associator (x, y, z) = (xy)z − x(yz).

These rings are a generalization of alternative rings.In this paperwe

prove that a prime antiflexible derivation alternator R is either as-

sociative or the nucleus is equal to the center of R. Also , a prime

antiflexible derivation alternator ring R with idempotent e not equal

to 1 and characteristic not equal to 2,3 is alternative.

Jordan Derivation On Prime Γ -Rings

S.Petchimuthu

Periyar University, Salem

In this paper we prove that the existence of a non zero Jordan deriva-

tion d such that [uαd(u), v]β = 0 , [uαuβd(uαuαu), v]β = 0, xαyβu =

xβyαu and d(uaa) = uαd(a) for all u, v ∈ U, α, β, δ ∈ Γ, x, y ∈ M

and a ∈ M − U where U is non zero left ideal of M on 2-torsion-

free prime Γ -ring M having no zero division forces the Γ − ring

M to be commutative. We also prove that the existence of Jordan

derivations d and g on prime Γ− ring M having no zero divi-

sors such that d 6= g, d(u)αvβd(w) = −g(u)αvβg(w), d(u)αvβg(w) =

29

−g(u)αvβd(w), d(uαa) = uαd(a), g(uαa) = uαg(a) and xαyβu =

xβyαu for all u, v, w ∈ U, aα, β ∈ Γ, x, y ∈ M and a ∈ M −U where

U is non zero left ideal of M forces the Γ− ring M to be commutative.

Rough Ideals of Nearrings

Kuncham Syam Prasad

Manipal Institute of Technology, Manipal, India

(This is a joint work with Bhavanari Satyanarayana,Kedukodi

Babushri Srinivas)

We introduce the notions of lower and upper approximations

for the ideal of a right near-ring N. on N and obtain fundamental

properties of these approximations. We prove that the lower and upper

approximations are the ideals of N. We define a congruence relation

We find that if an ideal has the insertion of factors property then

corresponding rough ideal has the same. Finally, we discuss rough

prime ideals of near-rings.

Some generalisations of differential operators in rings and

algebras

M. A. Quadri

Aligarh Muslim University, Aligarh - 202002 (INDIA)

The differential operator D possesses two basic properties namely,

(i) D is linear (ii) D satisfies Liebnitz rule

This motivated algebraists to define various linear maps in rings

which preserve some aspects of multiplicative structures such as deriva-

tions, Jordan homomorphisms and left derivations etc. During the

past few years, these maps are thoroughly studied and extended to

30

spaces and algebras. In the present article, we are principally inter-

ested in certain generalisations on Banach algebras that explore their

their continuity and structures of the algebras.

On Γ -Derivations Of Semiprime Γ -Near-Rings

Ravi Raina

SMVD University, Jammu and Kashmir

Throughout this paper M denotes right Γ -near-ring. A Γ -near-ring

is a triple (M, +, Γ) which satisfies the following conditions.

1. (M, +) is a group.

2. Γ is a non-empty set of binary operators on M such that for

each γ ∈ Γ ; (M, +, γ) is a near-ring.

3. xβ(yγz) = (xβy)γz for all x, y, z ∈ M and β, γ ∈ Γ .

M is called a prime Γ - near-ring if xΓMΓy = {0} implies x = 0 or

y = 0 for x, y ∈ M and γ ∈ Γ . M is called semiprime Γ - near-ring if

xΓMΓx = {0} implies x = 0 for x ∈ M and γ ∈ Γ . A Γ -derivation

on M is an additive endomorphism δ on M satisfying the product

rule δ(xγy) = δ(x)γy + xγδ(y) for all x, y ∈ M and γ ∈ Γ .

If M and N are Γ− near-rings then the mapping f : M → N

is called a Γ− near-ring homomorphism if f(x + y) = f(x) + f(y)

and f(xγy) = f(x)γf(y) and anti- Γ− near-ring homomorphism if

f(xγy) = f(y)γf(x) .

In the main Theorem of [1], Nurcan Agrac proved that if N is

a semiprime right near-ring, δ a derivation on N and A a subset of

N such that 0 ∈ A , AN ⊆ A and δ acts as a homomorphism on A

or as an anti-homomorphism on A , then δ(A) = 0 .

31

We generalize the above result for Γ− near-rings and prove that

if δ is a Γ -derivation on a semiprime right Γ− near-ring M and

A a subset of M such that 0 ∈ A , AΓM ⊆ A and δ acts as a

homomorphism on A or as an anti-homomorphism on A , then δ(A) =

0 .

1. Argac, Nurcan; On Prime and Semiprime Near-rings with Deriva-

tions, Internat. J. Math. Sci. vol. 20 No. 4 (1997) 737-740.

Certain conditioned rings

Rekha Rani

Aligarh Muslim University, Aligarh

Using commutativity of rings satisfying (xy)n(x,y) = xy proved

by Searcoid and MacHale [Amer. Math. Monthly, 93(1986), 121 122]

and S. Ligh and J. Luh [Amer. Math. Monthly, 93(1989), 40 41]

have given a direct sum decomposition for rings with the mentioned

condition. Further, Bell and Ligh [Math. J. Okayama Univ. 31 (1989),

93 99] sharpened the result and obtained decomposition theorems for

rings. In the present paper, we continue the study and investigate the

structure of certain rings satisfying either of the conditions:

(i)xy = ymxnp(x, y)or(i)xy = xmynp(x, y), where p(x, y) ∈ Z(x, y),

the ring of polynomials in two noncommuting indeterminates over the

ring Z of integers.

Infact we prove :

Theorem : Let R be a ring satisfying either of the conditions (i)

and (ii), then R = P⊕

N , where P is the set of all potent elements

of R and N is the set of all nilpotent elements of R .

32

On right multiplier in rings

Shakir Ali

Aligarh Muslim University, Aligarh (U.P.), India

Let R be an associative ring. An additive mapping H : R −→ R

is called a right(resp.left) multiplier if H(xy) = xH(y)(resp.H(xy) =

H(x)y), holds for all x, y ∈ R . A multiplier is an additive mapping

which is both right as well as left multiplier.

There has been a great deal of work concerning relationship be-

tween the commuatativity of a ring R and the exitence of certain spe-

cific types of derivations of R . Recently, many authors viz. Ashraf and

Nadeem [East-West J. Math 3(1)(2001), 87-91], Bell and Martindale

[Canad. Math. Bull. 30(1987), 92-101], Bresar[ J. Algebra 156 (1993),

385-394] and Hongan [Internat. J. Math. & Math. Sci. 2 (1997), 413-

415] have obtained commutativity of prime and semiprime rings with

derivation involving certain polynomials constrants. Very recently,

author together with Ashraf and Asma [Southeast Asian Bull. Math.

30(2006), 1-7)]established that a prime ring R with a non-zero ideal I

must be commutative if it admits a generalized derivation F satisfying

either of the properties: (i) F (xy)± xy ∈ Z(R) , (ii) F (xy)± yx ∈Z(R) , for all x, y ∈ R .

The aim of this paper is to establish the comutativity of R ad-

mitting a nonzero right multiplier. In fact, we prove the following

results:

Theorem. Let R be a prime ring and I be a nonzero ideal of R . If

R admits a nonzero right multiplier H satisfying any one of the above

properties, then R is commutative or H(I) = I (resp. H(I) = −I) .

Some related results have also been obtained for right multipliers.

33

On the centralizer algebra of the wreath product

(Z2 × Z2) o Sn

B.Sivakumar

University of Madras, Chennai, India

(This is a joint work with M.Parvathi)

We give a complete set of inequivalent irreducible representations

of the wreath product (Z2 × Z2) o Sn by using the Bratelli diagram

arising out of partitions of 4n whose 4 -core is empty.

We also study the centralizer algebra of (Z2×Z2) oSn acting on

a permutation module of S2n which gives rise to a new class of subal-

gebras of the edge colored partition algebras having a basis consisting

of vertex colored minimal diagrams having even number of vertices in

each class.

Application of Equiprime Fuzzy Ideals to Roughness in

Rings

Kedukodi Babushri Srinivas

Manipal Institute of Technology, Karnataka, India

(This is a joint work with Satyanarayana Bhavanari,Syam Prasad

Kuncham)

We recently introduced the notion of an equiprime fuzzy ideal

of a nearring and studied the radicals of fuzzy ideals. The concept is

significant because the cardinality of the image of an equiprime fuzzy

ideal can be greater than 2, which is unlike the fuzzy prime ideals of

rings and that of nearrings. In this paper, we provide the definitions

of lower approximation and upper approximation of a subset of a ring

with respect to a fuzzy ideal and a reference point. This expands

the application domain of the rough set model in rings based on fuzzy

ideals. Then we apply the concept of equiprime fuzzy ideal to compare

34

the lower and upper approximations of a set with that of the existing

model.

Brown Mc-Coy radicals for near-rings

Ravi Srinivasa Rao

P.B. Siddhartha College of Arts and Science, Vijayawada, A.P.

(This is a joint work with T. Sujatha)

In this paper two more radicals B s1 and B s

1(0) are introduced for

near-rings which generalize the Brown-McCoy radical of rings. It is

proved that B s1 is an ideal-hereditary Kurosh-Amitsur radical (KA-

radical) in the class of all zero-symmetric near-rings but it is not a

KA-radical in the class of all near-rings. Moreover B s1(0) is a KA-

radical in the class of all near-rings and for a near-ring R, B s1(0) (I) ⊆

B s1(0) (R) ∩ I for all ideals I of R and the equality holds if I is a left

invariant ideal of R.

On Generalized Right Alternator Rings

K.Suvarna

Sri Krishnadevaraya University, Anantapur, A.P.

(This is a joint work with C.Jaya Subba Reddy)

A generalized right alternative ring is a non-associative ring R

satisfying the identities:

(xy, z, w) + (x, y, (z, w)) = x(y, z, w) + (x, z, w)y,

(x, x, x) = 0, for all x, y, z, w in R.

R is said to be semiprime if for any ideal A of R , square of A = 0

implies A = 0 . In this paper we show that the additive subgroup I

spanned by all associators of the form (x, y, y) where x and y range

over a generalized right alternative ring R is a two-sided ideal of R .

35

Using this we prove that a semiprime generalized right alternative ring

is a right alternative.

R-S Correspondence for G -Brauer algebras

A.Tamilselvi

University of Madras, Chennai, India

(This is a joint work with M.Parvathi)

In this paper, we establish the Robinson Schensted correspon-

dence for the wreath product GoSn where G is cyclic, which gives the

bijection between the elements of G oSn and pairs of standard tableau

of the same shape λ `a qn . Our approach is different from that of

Shimozono, White [ShW] and Stanton, White [StW]. As a biprod-

uct, the Robinson Schensted correspondence for G -Brauer algebras

is obtained, which gives the bijection between the set of G -Brauer

diagrams En and the set of pairs of G -vacillating tableau of shape

λ ∈ Γn, where Γn = {λ | λ `a q(n−2k) where k is an integer with 0 ≤k ≤ n

2} .

36