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On 2-absorbing Commutative Semigroups
On 2-absorbing Commutative Semigroups
Ahmad Yousefian Darani1
(Joint with: Edmund Puczylowski2)
1Department of Mathematics and ApplicationsUniversity of Mohaghegh Ardabili
2Department of MathematicsUniversity of Warsaw
Groups and Their ActionsJune 22-26, 2015, Bedlewo, Poland
On 2-absorbing Commutative Semigroups
History
2-absorbing ideals (Ayman Badawi, 2007)
A commutative ring R is called 2-absorbing if, for arbitrary elementsr1, r2, r3 of R such that r1r2r3 = 0 there are 1 ≤ i 6= j ≤ 3 for whichri rj = 0.
On 2-absorbing Commutative Semigroups
History
.
Clearly 2-absorbing rings generalize in a natural way prime rings.
Badawi described the structure of such rings and using it it wasshown that a ring R is 2-absorbing if and only if for arbitraryideals I1, I2, I3 of R such that I1I2I3 = 0 there are 1 ≤ i 6= j ≤ 3for which Ii Ij = 0. Thus 2-absorbing rings can be defined intwo equivalent ways, by elements or by ideals.
It became even more interesting when we observed that forgraded rings the respective notions defined in terms of homoge-neous elements and homogeneous ideals are already not equiv-alent.
Trying to explain that phenomena we came to a conclusion thata more appropriate context for these studies form commutative(multiplicative) semigroups with 0.
On 2-absorbing Commutative Semigroups
Contract
Throughout this talk S denotes a (multiplicative) commutativesemigroup with 0.
In what follows B denotes the prime radical of S .
For arbitrary elements x , y ∈ S we denote by (x) and (x , y) theideals of S generated by x and x , y , respectively.
For a given subset X of S we denote by ann(X ) = {s ∈ S |Xs = 0}.
On 2-absorbing Commutative Semigroups
Definition
.
The definition of 2-absorbing rings as well as their ideal characteriza-tion are given in terms of multiplication only, so they can be extendto commutative (multiplicative) semigroups with 0.
2-absorbing semigroups
We say that S is 2-absorbing if, for arbitrary elements s1, s2, s3 ∈ Ssatisfying s1s2s3 = 0, there are 1 ≤ i 6= j ≤ 3 such that si sj = 0.
Strongly 2-absorbing semigroups
We say that S is strongly 2-absorbing if, for arbitrary ideals I1, I2, I3of S satisfying I1I2I3 = 0, there are 1 ≤ i 6= j ≤ 3 such that Ii Ij = 0.
On 2-absorbing Commutative Semigroups
Counterexample
It is clear that if S is strongly 2-absorbing, then it is 2-absorbing.The following simple example shows that the converse does not hold.
Example
Let S be the Rees factor of the free commutative semigroup with 0generated by x , y modulo the ideal generated by x2, y2. It is easy tosee that S3 = 0 but S2 6= 0. Hence S is not strongly 2-absorbing. Ifs1, s2, s3 are images in S of words w1,w2,w3 of the free semigroup,then two of them, say, w1,w2 contain x or y . Hence s1s2 = 0, whichshows that S is 2-absorbing.
On 2-absorbing Commutative Semigroups
Strategy
We start our studies with describing the structure of strongly2-absorbing semigroups.
We apply them next to rings and rings graded by abelian groups.
It is clear that they can be applied to many other ring (or evensemiring) situations.
We show also that the multiplicative semigroups of commu-tative rings are 2-absorbing if and only they are strongly 2-absorbing.
On 2-absorbing Commutative Semigroups
Some Characterizations of strongly2-absorbing semigroups
On 2-absorbing Commutative Semigroups
The First Characterization of strongly 2-absorbingsemigroups
The following characterization of strongly 2-absorbing semigroups isquite useful as makes it possible to apply in some studies inductionarguments.
.
S is strongly 2-absorbing if and only if for arbitrary ideals F1,F2,F3generated by ≤ 2 elements and satisfying F1F2F3 = 0, there are1 ≤ i 6= j ≤ 3 such that FiFj = 0.
On 2-absorbing Commutative Semigroups
The Second Characterization of strongly 2-absorbingsemigroups
The semigroup S is strongly 2-absorbing if and only if
(a) If I , J are ideals of S such that I 6⊆ B, J 6⊆ B and IJ ⊆ B, thenIJ = 0 and IB = 0 = JB;
(b) For every subset X of B, ann(X ) is a prime ideal of S ;
(c) If I , J,K are ideals of S not contained in B, then IJ 6= 0 orJK 6= 0 or IK 6= 0.
On 2-absorbing Commutative Semigroups
The Third Characterization of strongly 2-absorbingsemigroups
The semigroup S is strongly 2-absorbing if and only if
(1) for every subset X of B, ann(X ) is a prime ideal of R;
(2) one of the following conditions holdsa) B is a prime ideal of Sb) S contains prime ideals P1,P2 such that P1P2 = 0.
On 2-absorbing Commutative Semigroups
Applications to Rings
Applications to Rings
In what follows R is a commutative ring and S(R) denotes themultiplicative semigroup of R.
On 2-absorbing Commutative Semigroups
Applications to Rings
Theorem
The following conditions are equivalent
(a) R is 2-absorbing;
(b) S(R) is 2-absorbing;
(c) S(R) is strongly 2-absorbing;
(d) for arbitrary ideals I1, I2, I3 of R satisfying I1I2I3 = 0 there are1 ≤ i 6= j ≤ 3 such that Ii Ij = 0.
On 2-absorbing Commutative Semigroups
Applications to Rings
Graded rings
Let G be an abelian group and let R be a G -graded ring. Recallthat R =
⊕g∈G Rn, the direct sum of additive subgroups Rg
of R, with RgRh ⊆ Rgh for all g , h ∈ G .
An ideal I of R is called homogeneous if I =⊕
g∈G (I ∩ Rg ).
Denote by Sh(R) the multiplicative semigroup of homogeneouselements of R.
On 2-absorbing Commutative Semigroups
Applications to Rings
Theorem
For a given G -graded ring R the following conditions are equivalent
(a) Sh(R) is 2-absorbing;
(b) For arbitrary homogeneous ideals I1, I2, I3 satisfying I1I2I3 = 0there are 1 ≤ i 6= j ≤ 3 such that Ii Ij = 0;
(c) For every X ⊆ B consisting of homogeneous elements, ann(X )is a graded prime ideal and one of the following conditions holds(i) Bh is a graded prime ideal of R;(ii) R contains graded prime ideals P1,P2 such that P1P2 = 0.
On 2-absorbing Commutative Semigroups
Applications to Rings
Example
If R is a G -graded 2-absorbing ring then, by the previous Theorem,Sh(R) is 2-absorbing. The converse does not hold. For instance ifR is a finite abelian group of order ≥ 3 and F is an algebaricallyclosed field of characteristic 0 and R is the group algebra of G overF , then obviously Sh(R) is 2-absorbing but, since R is isomorphicto the direct sum of | G | copies of F , R is not 2-absorbing.
The following result shows that the converse holds if G is torsion-free.
On 2-absorbing Commutative Semigroups
Applications to Rings
Example
If R is a G -graded 2-absorbing ring then, by the previous Theorem,Sh(R) is 2-absorbing. The converse does not hold. For instance ifR is a finite abelian group of order ≥ 3 and F is an algebaricallyclosed field of characteristic 0 and R is the group algebra of G overF , then obviously Sh(R) is 2-absorbing but, since R is isomorphicto the direct sum of | G | copies of F , R is not 2-absorbing.
The following result shows that the converse holds if G is torsion-free.
On 2-absorbing Commutative Semigroups
Applications to Rings
Theorem
Suppose that R is a G -graded ring and G is torsion-free. If Sh(R)is strongly 2-absorbing, then R is 2-absorbing.
On 2-absorbing Commutative Semigroups
Applications to Rings
Example
The following example shows that the previous Theorem need nothold if Sh(R) is a 2-absorbing ring.
Example. Let F be a field and A = F [x , y ]/I , where I is the idealof F [x , y ] generated by x2, y2. Set a = x + I and b = y + I . TheF -subalgebra R = F + Faz + Fbz2 + Fabz3 of A[z ] is graded ina canonical way by the additive group of integers. Note that fort = az + bz2 we have t3 = 0 and t2 6= 0, so R is not 2-absorbing.However every non-invertible element of Sh(R) is square-zero, soSh(R) is 2-absorbing.
On 2-absorbing Commutative Semigroups
Applications to Rings
Example
The following example shows that the previous Theorem need nothold if Sh(R) is a 2-absorbing ring.
Example. Let F be a field and A = F [x , y ]/I , where I is the idealof F [x , y ] generated by x2, y2. Set a = x + I and b = y + I . TheF -subalgebra R = F + Faz + Fbz2 + Fabz3 of A[z ] is graded ina canonical way by the additive group of integers. Note that fort = az + bz2 we have t3 = 0 and t2 6= 0, so R is not 2-absorbing.However every non-invertible element of Sh(R) is square-zero, soSh(R) is 2-absorbing.
On 2-absorbing Commutative Semigroups
On n-absorbing and strongly n-absorbing rings
n-absorbing and strongly n-absorbing rings
On 2-absorbing Commutative Semigroups
On n-absorbing and strongly n-absorbing rings
Notation
A finite number of ideals of R (some ideals can appear several times)will be called a collection of ideals if their product is equal 0.
On 2-absorbing Commutative Semigroups
On n-absorbing and strongly n-absorbing rings
n-absorbing and strongly n-absorbing rings
Let n be an integer ≥ 2. A ring R is called strongly n − absorbingif the following condition is satisfied:(*) every collection of n + 1 ideals of R contains a collection of nideals.
If the condition (*) is satisfied for collections of n+1 principal ideals,then R is called n − absorbing .
On 2-absorbing Commutative Semigroups
On n-absorbing and strongly n-absorbing rings
n-absorbing and strongly n-absorbing rings
Let n be an integer ≥ 2. A ring R is called strongly n − absorbingif the following condition is satisfied:(*) every collection of n + 1 ideals of R contains a collection of nideals.If the condition (*) is satisfied for collections of n+1 principal ideals,then R is called n − absorbing .
On 2-absorbing Commutative Semigroups
On n-absorbing and strongly n-absorbing rings
n-absorbing and strongly n-absorbing rings
Let n be an integer ≥ 2. A ring R is called strongly n − absorbingif the following condition is satisfied:(*) every collection of n + 1 ideals of R contains a collection of nideals.If the condition (*) is satisfied for collections of n+1 principal ideals,then R is called n − absorbing .
On 2-absorbing Commutative Semigroups
On n-absorbing and strongly n-absorbing rings
A Conjecture by Badawi and Anderson
Question
It is evident that strongly n-absorbing rings are n-absorbing. Badawiand Anderson asked whether the converse holds as well. We havean answer to this question as follows:
Theorem
If R is n-absorbing and the additive group of R is torsion-free, thenR is strongly n-absorbing.
On 2-absorbing Commutative Semigroups
On n-absorbing and strongly n-absorbing rings
References
D. F. Anderson and A. Badawi, On n-absorbing ideals ofcommutative rings, Comm. Algebra 39 (2011), 1646–1672
Ayman Badawi, On 2-absorbing ideals of commutative rings,Bull. Austral. Math. Soc. 75(2007), 417–429.
L. Fuchs, Infinite abelian groups. Vol. I. Pure and AppliedMathematics, Vol. 36 Academic Press, New York-London 1970
C. Nastasescu and F. Van Oystaeyen, Graded ring theory,Noth-Holland, Amsterdam 1982.