On 2-Absorbing Commutative Semigroups

On 2-absorbing Commutative Semigroups

Ahmad Youseﬁan Darani1 (Joint with: Edmund Puczylowski2)

1Department of Mathematics and Applications University of Mohaghegh Ardabili

2Department of Mathematics University of Warsaw

Groups and Their Actions June 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 elements r1, r2, r3 of R such that r1r2r3 = 0 there are 1 ≤ i 6= j ≤ 3 for which ri 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 was shown that a ring R is 2-absorbing if and only if for arbitrary ideals I1, I2, I3 of R such that I1I2I3 = 0 there are 1 ≤ i 6= j ≤ 3 for which Ii Ij = 0. Thus 2-absorbing rings can be deﬁned in two equivalent ways, by elements or by ideals. It became even more interesting when we observed that for graded rings the respective notions deﬁned in terms of homogeneous elements and homogeneous ideals are already not equivalent. Trying to explain that phenomena we came to a conclusion that a 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) commutative semigroup 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) the ideals 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 characterization are given in terms of multiplication only, so they can be extend to commutative (multiplicative) semigroups with 0.

2-absorbing semigroups

We say that S is 2-absorbing if, for arbitrary elements s1, s2, s3 ∈ S satisfying 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, I3 of 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 0 generated by x, y modulo the ideal generated by x2, y 2. It is easy to see that S3 = 0 but S2 6= 0. Hence S is not strongly 2-absorbing. If s1, 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, which shows that S is 2-absorbing. On 2-absorbing Commutative Semigroups

Strategy

We start our studies with describing the structure of strongly 2-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 even semiring) situations. We show also that the multiplicative semigroups of commutative rings are 2-absorbing if and only they are strongly 2- absorbing. On 2-absorbing Commutative Semigroups

Some Characterizations of strongly 2-absorbing semigroups On 2-absorbing Commutative Semigroups

The First Characterization of strongly 2-absorbing semigroups

The following characterization of strongly 2-absorbing semigroups is quite useful as makes it possible to apply in some studies induction arguments. .

S is strongly 2-absorbing if and only if for arbitrary ideals F1, F2, F3 generated by ≤ 2 elements and satisfying F1F2F3 = 0, there are 1 ≤ i 6= j ≤ 3 such that Fi Fj = 0. On 2-absorbing Commutative Semigroups

The Second Characterization of strongly 2-absorbing semigroups

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, then IJ = 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 or JK 6= 0 or IK 6= 0. On 2-absorbing Commutative Semigroups

The Third Characterization of strongly 2-absorbing semigroups

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 holds a) B is a prime ideal of S b) 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 the multiplicative 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 are 1 ≤ 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. Recall L that R = g∈G Rn, the direct sum of additive subgroups Rg of R, with Rg Rh ⊆ Rgh for all g, h ∈ G. L An ideal I of R is called homogeneous if I = g∈G (I ∩ Rg ). Denote by Sh(R) the multiplicative semigroup of homogeneous elements 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 = 0 there 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 if R is a ﬁnite abelian group of order ≥ 3 and F is an algebarically closed ﬁeld of characteristic 0 and R is the group algebra of G over F , then obviously Sh(R) is 2-absorbing but, since R is isomorphic to 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

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 not hold if Sh(R) is a 2-absorbing ring.

Example. Let F be a ﬁeld and A = F [x, y]/I , where I is the ideal of F [x, y] generated by x2, y 2. Set a = x + I and b = y + I . The F -subalgebra R = F + Faz + Fbz2 + Fabz3 of A[z] is graded in a canonical way by the additive group of integers. Note that for t = 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, so Sh(R) is 2-absorbing. On 2-absorbing Commutative Semigroups Applications to Rings Example

The following example shows that the previous Theorem need not hold if Sh(R) is a 2-absorbing ring.

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. 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 − absorbing if the following condition is satisﬁed: (*) every collection of n + 1 ideals of R contains a collection of n ideals. 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 − absorbing if the following condition is satisﬁed: (*) every collection of n + 1 ideals of R contains a collection of n ideals. 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

Question It is evident that strongly n-absorbing rings are n-absorbing. Badawi and Anderson asked whether the converse holds as well. We have an answer to this question as follows:

Theorem If R is n-absorbing and the additive group of R is torsion-free, then R 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 of commutative 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 Applied Mathematics, Vol. 36 Academic Press, New York-London 1970 C. Nastasescu and F. Van Oystaeyen, Graded ring theory, Noth-Holland, Amsterdam 1982.