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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

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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

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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.

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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.