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### Article

Kyungpook Mathematical Journal 2022; 62(3): 425-436

Published online September 30, 2022

### On 2-absorbing Primary Ideals of Commutative Semigroups

Manasi Mandal∗ and Biswaranjan Khanra

Department of Mathematics, Jadavpur University, Kolkata-700032, India
e-mail : manasi_ju@yahoo.in and biswaranjanmath91@gmail.com

Received: January 5, 2021; Revised: April 24, 2022; Accepted: May 3, 2022

### Abstract

In this paper we introduce the notion of 2-absorbing primary ideals of a commutative semigroup. We establish the relations between 2-absorbing primary ideals and prime, maximal, semiprimary and 2-absorbing ideals. We obtain various characterization theorems for commutative semigroups in which 2-absorbing primary ideals are prime, maximal, semiprimary and 2-absorbing ideals. We also study some other important properties of 2-absorbing primary ideals of a commutative semigroup.

Keywords: Commutative semigroup, Prime ideal, Maximal ideal, Semiprime ideal, Semiprimary ideal, 2-absorbing ideal

### 1. Introduction

The concept of a 2-absorbing ideal for a commutative ring was introduced by Badawi [2] and later extended to commutative semigroups by Cay et al. [4] as follows : a proper ideal I of a commutative semigroup S is said to be a 2-absorbing ideal if abc∈ I implies either ab∈ I or ac∈ I or bc∈ I for some a,b,c∈ S. The notion of a 2-absorbing primary ideal of a commutative ring was introduced as a generelisation of a 2-absorbing ideal by Badawi [3]. Studies of commutative algebraic structures (rings, semirings) via 2-absorbing primary ideals have been made by many authors in ([3],[8],[10]) and 2-absorbing primary ideals in lattices were defined and studied in [15]. The main purpose of this paper is to define the concept of 2-absorbing primary ideal, and to invetigate how this notion can be used in the study of commutative semigroups.

In this paper, we define a 2-absorbing primary ideal in a commutative semigroup (cf. Definition 2.1). Clearly 2-absorbing ideals, prime ideals and maximal ideals are 2-absorbing primay ideals (cf. Theorem 2.2, Corollary 2.6) and we prove that every semiprimary ideal of S is a 2-absorbing primay ideal (cf. Theorem 2.14). The converses is not true (cf. Remark 2.7, 2.8, 2.16). Then we characterize the semigroups for which all 2-absorbing primary ideals are prime, maximal, 2-absorbing and semiprimary (cf. Theorem 2.10, 2.12, 2.9, 2.17,), as a result we obtain that a semigroup in which 2-absorbing primary ideals are semiprimary is equivalent to a semiprimary semigroup (cf. Theorem 2.17) and semigroup in which 2-absorbing primary ideals are maximal is either a group or union of two groups (cf. Corollary 2.13). Then we prove that a proper ideal I of a semigroup S is 2-absorbing primary ideal of S if and only if I[x] (I[x1,x2,,xn]) is a 2-absorbing primary ideal of the polynomial semigroup S[x] (S[x1,x2,,xn]) (cf. Theorem 2.31) and for any ideal I of S, I is a 2-absorbing primary ideal of S if and only if I is a 2-absorbing ideal of S (cf. Theorem 2.25). As a consequence we prove that the radical of a 2-absorbing primary ideal I is a 2-absorbing ideal, moreover if I=P is a prime ideal of S, then the residual (I:x)={sS:sxI} of I by xSI is a 2-absorbing primary ideal with (I:x)=P (cf. Theorem 2.24). We prove that the arbitrary union of 2-absorbing primary ideals is 2-absorbing primary but that intersections of 2-absorbing primary ideals need not be a 2-absorbing primary ideal (cf. Example 2.26). Also we find equivalence classes in the semigroup of all 2-absorbing primary ideals of a semigroup S and each class is closed under finite intersections (cf. Theorem 2.27). We observe that under certain conditions 2-absorbing primary ideals remains invariant under homomorphism of semigroups and it's inverse mapping (cf. Theorem 2.35). Lastly we also study 2-absorbing primary ideals in direct product of semigrops (cf. Theorem 2.38, 2.39) .

Before going to the main work we discuss some preliminaries which are necessary:

Definition 1.1. ([13]) A non-empty ideal P of a semigroup S is said to be prime if IJP implies that IP or JP, I, J being ideals of S. An ideal P is said to be completely prime if ab ∈ P implies either a ∈ P or b ∈ P for some a,b ∈ S.

Remark 1.2. These concepts coincide if S is commutative.

Definition 1.3. ([11]) For an ideal I of a semigroup S, radical of I, is defined as I={xS:xnI for some natural numbers n}, is the intersections of all prime ideals containing I.

Definition 1.4. ([9]) An ideal I of a semigroup S is called primary if ab∈ I implies either a∈ I or bI. An ideal I of a semigroup S is called semiprimary if I is a prime ideal of S.

Definition 1.5. ([11]) A commutative semigroup S is said to be fully prime if every ideal of S is prime and primary if every ideal of S is a primary ideal of S.

Definition 1.6. ([9]) A commutative semigroup S is said to be semiprimary semigroup if every ideal of S is semiprimary. Moreover, S is a semiprimary semigroup if and only if prime ideals of S are linearly ordered.

Definition 1.7. ([13]) An ideal M of a semigroup S is called a maximal ideal of S if MS and there does not exist an ideal M1 of S such that MM1S. If S is a semigroup with unity then S has unique maximal ideal, which is the union of all proper ideals of S, is prime also.

Theorem 1.8. ([11]) If I and J are any two ideals of a commutative semigroup S, then the following statements are true

(1) IJIJ.

(2) II=I.

(3) IJIJ.

(4) IJ=IJ=IJ. (5) (IJ)K=IKJK, where K is an ideal of S.

(5) If A is a prime ideal of S, then A=A and if A is a primary ideal of S, then A is a prime ideal of S.

Definition 1.9.([13]) An ideal I of a semigroup S is said to be a semiprime ideal if a2I for some a∈ S implies a∈ I.

Definition 1.10. ([5]) Let S be a semigroup and x be an indeterminate. Then S[x]={sxi:sS,i0} forms a semigroup wiith respect to the multiplication defined as : (sxi)(txj)=(st)xi+j, where s,t ∈ S and i,j ≥ 0, called the polynomial semigroup over S. Similarly we can define the polynomial semigroup S[x1,x2,,xn] in n variables.

### 2. Some Properties of 2-absorbing Primary Ideals

Throughout this paper, unless otherwise mentioned, S stands for a commutative semigroup.

Definition 2.1. A proper ideal I of a commutative semigroup S is said to be 2-absorbing primary if abc∈ I implies either ab ∈ I or bcI or acI for some a,b,c ∈ S.

Since II for any ideal I of a semigroup S so we have the following result

Theorem 2.2. Let S be a commutative semigroup. Then every 2-absorbing ideal of S is a 2-absorbing primary ideal of S.

The following lemmas are obvious, hence we omit the proof.

Lemma 2.3.([Lemma 2.1, [6]]) Let S be a commutative semigroup. Then every prime ideal of S is a 2-absorbing ideal of S.

Lemma 2.4.([Theorem 2.4, [6]]) Let S be a commutative semigroup. Then every maximal ideal of S is a 2-absorbing ideal of S.

Lemma 2.5.([Lemma 2.9, [6]]) Let P1 and P2 be two prime ideals of a semigroup S. Then P1 ∩ P2 is a 2-absorbing ideal of S.

Corollary 2.6. Let S be a commutative semigroup. Then

(1) if P1 and P2 are two prime ideals of S then P1P2 is a 2-absorbing primary ideal of S.

(2) every maximal ideal of S is a 2-absorbing primary ideal of S.

(3) every prime ideal of S is a 2-absorbing primary ideal of S.

Remark 2.7. The following example shows that converse of Lemma 2.3. and 2.4. are not true. Consider the ideal I2={n:n2} in the semigroup S=({0},+), which is 2-absorbing primary (as well 2-absorbing) but neither prime nor a maximal ideal of S.

Remark 2.8. The following example shows that converse of Theorem 2.2 is not true. Consider the ideal I=(m:m6) in the semigroup S=(,+). Then 1+2+3 ∈ I but neither 1+2 ∈ I nor 2+3 ∈ I nor 1+3 ∈ I. Clearly, I is a 2-absorbing primary ideal of S but not a 2-absorbing ideal of S.

A semigroup S is called regular if for each element s ∈ S there exists an element x ∈ S such that sxs=s ([7], Section 5). Since in a commutative regular semigroup every ideal coincide with its radical ([7], Theorem 5.1), we have the following result.

Corollary 2.9. Let S be a commutative regular semigroup. Then an ideal I of S is a 2-absorbing primary ideal of S if and only if I is a 2-absorbing ideal of S.

The following is a characterization of a semigroup in which 2-absorbing primary ideals are prime:

Theorem 2.10. Let S be a commutative semigroup. Then every 2-absorbing primary ideals of S are prime if and only if prime ideals of S are linearly ordered and A=A for every 2-absorbing primary ideal A of S.

Proof. Let P1 and P2 be two prime ideals of S. Then P1P2 is a 2-absorbing primary ideal of S (cf. Corollary 2.6(1)) and so prime by hypothesis. Hence prime ideals are linearly ordered. Again let A be a 2-absorbing primary ideal of S and so prime ideal of S. Therefore A=A.

Conversely, Let A be a 2-absorbing primary ideal of S. Since prime ideals are linearly ordered so A=A= αΛPα=Pβ for some βΛ, where {Pα}αΛ are prime ideals of S containing A. Hence the result follows.

Since every primary ideals of a commutative semigroup S is 2-absorbing primary, we have the following result by using Theorem 3.1 of [12].

Corollary 2.11. Let S be a commutative semigroup with zero and identity in which nonzero 2-absorbing primary ideals are prime. Then S satisfies one of the following conditions.

(i) S=HM, where H is the group of units in S and M={0,ah:aM,a2=0,hH}.

(ii) Mn=M for every positive integer n.

The following is a characterization of a semigroup in which 2-absorbing primary ideals are maximal:

Theorem 2.12. Let S be a commutative semigroup with unity. Then 2-absorbing primary ideals of S are maximal if and only if S is either a group or has a unique 2-absorbing primary ideals A such that S=AH, where H is the group of units of S.

Proof. Let S be commutative semigroup with unity in which 2-absorbing primary ideals are maximal. If S is not a group, it has unique maximal ideal say A and since maximal ideals are 2-absorbing primary (cf. Corollary 2.6(2)) so has unique 2-absorbing primary ideal A. Therefore S=A ∪ H, where H is the group of units of S.

Conversely, if S is a group then it has no 2-absorbing primary ideal so the condition satisfied vacously. Again if S has unique 2-absorbing primary ideal then clearly it is maximal.

Moreover, we prove that A is also a group. Clearly A is the unique prime ideal of S. Then for any a ∈ A, aS=A. Hence aS is a 2-absorbing primary ideal of S. Hence aS=A for every a ∈ A, by hypothesis. Then aS=a2S=A implies a=a2xax=a2x2. Thus ax is an idempotent element of A. If possible let e, f be two idempotent element of S. Then eS=fSe=fe=ef=f. Consequently eS=aS=A. Therefore A is a group. So we can conclude the corollary

Corollary 2.13. Let S be a commutative semigroup with unity. Then 2-absorbing primary ideals are maximal if and only if either S is a group or S is a union of two groups.

Theorem 2.14. Let S be a commutative semigroup. Then every semiprimary ideal of S is a 2-absorbing primary ideal of S.

Proof. Let I be a semiprimary ideal of a semigroup S and abcII with abI for some a,b,cS. Hence I is a prime ideal of S.

Case(1). Suppose abI. Since I is a prime ideal of S so cI. Hence acI and bcI.

Case(2). Suppose abI. Since I is a prime ideal, we have either aI or bI. Hence either acI or bcI.

Therefore I is a 2-absorbing primary ideals of S.

The following are obvious consequence of above theorem:

Corollary 2.15. Let I be an ideal of a commutative semigroup S. Then

(1) if I is a prime ideal of S, then I is a 2-absorbing primary ideal of S.

(2) if I is a prime ideal of S, then In is a 2-absorbing primary ideal of S for each natural number n.

(3) every primary ideal of S is a 2-absorbing primary ideal of S.

Remark 2.16. The converse of Theorem 2.14 is not true. Consider the principal ideal I=(6) generated by 6 in the semigroup S={,.}, which is clearly 2-absorbing primary but I=(6) is not a prime ideal of S and hence not a semiprimary ideal of S.

The following theorem is a characterization of a semigroup in which 2-absorbing primary ideals are semiprimary:

Theorem 2.17. Let S be a commutative semigroup. Then the following statements are equivalent:

(1) 2-absorbing primary ideals of S are semiprimary.

(2) Prime ideals of S are linearly ordered.

(3) S is a semiprimary semigroup.

(4) Semiprime ideals are linearly ordered.

(5) Semiprime ideals of S are prime.

Proof. (1)(2) Let P1 and P2 be two prime ideals of S. Then P1P2 is a 2-absorbing primary ideal of S (cf. Corollary 2.6(1)) and so semiprimary ideal of S, by hypothesis. Therefore P1P2=P1P2=P1P2, is a prime ideal of S. Therefore either P1P2 or P2P1.

(2)(1) Since prime ideals of S are linearly ordered, then for any ideal I of S, I is a prime ideal of S. Consequently, 2-absorbing primary ideals of S are semiprimary.

(2)(3) follows from ([9], Theorem 1).

(2)(4) Let S1 and S2 are two distinct semiprime ideals of S. Then S1S2 is a semiprime ideal of S. Hence S1S2=S1S2, is a prime ideal of S, since prime ideals are linearly ordered. Hence semiprime ideals of S are linearly ordered.

(4)(2) It is clear.

(2)(5) Let I be a semiprime ideal of S. Then I=I, is a prime ideal of S.

(5)(2) Let P1 and P2 be two distinct prime ideals of S. Then P1P2 is a semiprime ideals of S, hence prime ideals of S. Consequently, prime ideals of S are linearly ordered.

Definition 2.18. A commutative semigroup S is said to be 2-absorbing primary if every proper ideal of S is a 2-absorbing primary ideal of S.

Example 2.19. Consider the commuttive semigroup S={,+}, which has no proper prime ideal. Clearly S is a primary semigroup and hence 2-absorbing primary semigroup.

Theorem 2.20. Let S be a commutative semigroup with unity. If

(i) proper prime ideals are maximal then S is a 2-absorbing primary semigroup.

(ii) 2-absobing ideals are semiprimary then S is a 2-absorbing primary semigroup.

Proof. (i) If S is commutative semigroup with unity in which proper prime ideals are maximal. Then S has unique proper prime as well as maximal ideal say M. Clearly for any ideal I of S, I=M, a prime ideal of S. Hence S is a 2-absorbing primary semigroup (cf. Corollary 2.15(1)).

(ii) Let P1 and P2 be two prime ideals of S. Then P1P2 is 2-absorbing ideal of S and hence semiprimary ideal of S. Then P1P2=P1P2=P1P2, is a prime ideal of S and hence prime ideals of S are linearly ordered. Then for any ideal I of S, I is a prime ideal of S and hence S is a 2-absorbing primary ideal of S (cf. Corollary 2.15(1)).

Theorem 2.21. Let S be a commutative semigroup. Then S can be written as disjoint union of 2-absorbing primary semigroup.

Proof. Let S be a commutative semigroup. Then S can be written as disjount union of archimedian subsemigroups of S ([12], Corollary 1.6). Clearly every archimedian semigroup is primary and every primary semigroup is 2-absorbing primary. Therefore S=Sα, where Sα is a 2-absorbing primary subsemigroup of S.

Theorem 2.22. Let S be a commutative regular semigroup. Then the following statements about S are equivalent:

(1) 2-absorbing primary ideals of S are semiprimary.

(2) prime ideals of S are linearly ordered.

(3) idempotents of S form a chain under natural ordering.

(4) All ideals of S are linearly ordered.

(5) S is a fully prime semigroup.

(6) S is a primary semigroup.

(7) S is a semiprimary semigroup.

(8) 2-absorbing ideals of S are prime.

Proof. (1) (2) follows from Theorem 2.17.

(2)(3)(4)(5)(6)(7) follows from ([9], Theorem 1).

(7) (8) Let I be a 2-absorbing ideal of S. Since S is semiprimary and regular so I=I=P, a prime ideal of S, as desired.

(8)(1) Let I be a 2-absorbing primary ideal of a regular semigroup S. Then I=I is a 2-absorbing ideal of S (cf. Theorem 2.24(1)) and hence prime by hypothesis. Therefore I is a semiprimary ideal of S, as desired.

Lemma 2.23. Let I be a 2-absorbing ideal of a semigroup S with unity. Then (I:x) is a 2-absorbing ideal of S for all x ∈ S-I.

Proof. Let abc ∈ (I:x) for some a,b,c ∈ S. Since x ∉ I, we have (I:x)=S. Then abcx ∈ I. Since I is a 2-absorbing ideal of S so either ab ∈ I or bcxI or acx ∈ I that is ab ∈ (I:x) or ac ∈ (I:x) or bc ∈ (I:x). Consequently (I:x) is a 2-absorbing ideal of S.

Theorem 2.24. Let I be a 2-absorbing primary ideal of a semigroup S with unity. Then the following statements are true

(1) I is a 2-absorbing ideal of S.

(2) If I is a prime ideal P of S then (I:x) is a 2-absorbing primary ideal of S with (I:x)=P for all xSI.

(3) If I is a maximal ideal M of S then (I:x) is a 2-absorbing primary ideal of S with (I:x)=M for all xSI.

(4) (I:x) is a 2-absorbing ideal of S for all xSI.

(5) (I:x)=(I:x2) for all xSI.

Proof. (1) Proof is similar to ([3], Theorem 2.2).

(2) Let xSI and p(I:x). Then pxII. Since I is a prime ideal of S and xI so pI. Hence I(I:x)I=P, which implies P=I(I:x)I=P. Consequently, (I:x)=P, a prime ideal of S and hence (I:x) is a 2-absorbing primary ideal of S (cf. Corollary 2.15.(1)).

(3) If I=M and maximal ideal of a semigroup with unity is prime, hence the proof follows from (2).

(4) Since I is a 2-absorbing primary ideal of S so I is a 2-absorbing ideal of S. Hence (I:x) is a 2-absorbing ideal of S (cf. Lemma 2.23).

(5) Let p(I:x2). Then x2pI and hence either x2I or xpI, since I is a 2-absorbing ideal of S. If x2I then xI, a contradiction. Hence xpI implies p(I:x). Therefore (I:x2)(I:x). Clearly, (I:x)(I:x2). Hence, (I:x)=(I:x2) for all xSI.

Theorem 2.25. Let I be an ideal of a semigroup S. Then I is a 2-absorbing primary ideal of S if and only if I is a 2-absorbing ideal of S.

Proof. Suppose I is a 2-absorbing primary ideal of S and abcI for some a,b,cS. So either abI or bcI=I or caI=I. Hence I is a 2-absorbing ideal of S.

Conversely, let I be a 2-absorbing ideal of S. Since 2-absorbing ideals are 2-absorbing primary (cf. Theorem 2.2) so clearly I is a 2-absorbing primary ideal of S.

Clearly arbitary union of 2-absorbing primary ideals of a semigroup S is 2-absorbing primary ideal but intersection of two 2-absorbing primary ideals need not be 2-absorbing primary ideal of S. We have the following example

Example 2.26. Let I1=5 and I2=6 be two ideals of the semigroup (,.). Clearly I1 and I2 are 2-absorbing primary ideal of S. Then I1I2=30, which is not a 2-absorbing ideal of S. Hence I1I2 is a not a 2-absorbing primary ideal of S (cf. Theorem 2.24(1)).

Let M be the set of all 2-absorbing primary ideals of a semigroup S and we define a relation ρ on M by I1ρI2 if and only if I1=A=I2 for some 2-absorbing ideal A of S. Clearly ρ is a congrurence on M. So every element of a ρ-equivalence class A is a A-2-absorbing primary ideal for some 2-absorbing ideal A of S. Clearly M forms a semigroup with respect to usual set union and each ρ-equivalence class of M is an element of the factor semigroup M/ρ.

Theorem 2.27. Each ρ-class of M is closed under finte intersections.

Proof. Let I1,I2,....In be elements of a ρ-class A for some 2-absorbing ideal A of S. Then (I1I2.....In)=A. Let abcI=Ii and ab ∉ I for some a,b,c ∈ S. Then ab ∉ Ii for some i{1,2,....,n}. Hence bcIi=A or acIi=A. Hence the result follows.

Therefore the semigroup M can be written as disjoint union of semigroups that is M={A:AM/ρ}.

Theoprerm 2.28. Let I1 and I2 be two P1-primary and P2- primary ideals for some prime ideals P1 and P2 of a commutative a semigroup S. Then I1I2 and I1I2 are 2-absorbing primary ideals of S.

Proof. The proof is similar to ([3], Theorem 2.4).

Proposition 2.29. Let I be a proper ideal of a semigroup S. Then the following statements are equivalent

(1). For every ideals J, K, L of S such that IJ, JKLI implies JKI or KLI or JLI.

(2). For every ideals J,K,L of S, JKLI implies either JKI or KLI or JLI.

Proof. (1)(2) Let JKLI for some ideals J, K, L of S. Then (JI)KL=JKLIKLI. Setting P=IJ we have IP and PKLI implies either PKI or PLI or KLI. Therefore either (JI)KI or (JI)LI or KLI implies JKI or KLI or JLI, as desired.

(2)(1) Straightforward.

Let S[x] be the polynomial semigroup of a semigroup S. Then if I is an ideal of S, then I[x]={axi:aI,i0} is an ideal of S[x] and also we have the following result:

Lemma 2.30. Let I be an ideal of a semigroup S. Then I[x]=I[x].

Proof. Let f(x)=axiI[x] for some a ∈ S and i≥ 0. Then for some n, anxinI[x]anIaIf(x)=axiI[x]. Therefore I[x]I[x].

Again let g(x)=bxiI[x]. Then bIbnI for some n. Hence (g(x))n=bnxinI[x]g(x)I[x]. Therefore I[x]I[x]. Consequently, I[x]=I[x].

Proposition 2.31. Let S be a commutative semigroup. Then a proper ideal I is a 2-absorbing primary ideal of S if and only if I[x] is a 2-absorbing primary ideal of S[x].

Proof.,Let I be a 2-absorbing primary ideal of S and (axi)(bxj)(cxk)I[x], where a,b,c ∈ S and i,j,k ≥ 0. Then abcxi+j+kI[x]abcIabI or acI or bcI, since I is a 2-absorbing primary ideal of S. Hence (axi)(bxj)=abxi+jI[x] or (axi)(cxj)=acxi+jI[x]=I[x] or (bxj)(cxk)=bcxj+kI[x]=I[x], since I[x]=I[x] (cf. Lemma 2.30). Therefore I[x] is a 2-absorbing primary ideal of S[x].

Conversely, let I[x] be a 2-absorbing primary of S[x] and abc ∈ I for some a,b,c ∈ S. Then (axi)(bxj)(cxk)=abcxi+j+kI[x], for some i,j,k0. Since I[x]=I[x] (cf. Lemma 2,30) so ab∈ I or acI or bcI. Therefore I is a 2-absorbing primary ideal of S.

The following result is a simple consequence of Proposition 2.31:

Corollary 2.32. Let I be a proper ideal of S. Then I is a 2-absorbing primary ideal of S if and only if I[x1,x2,,xn] is a 2-absorbing primary ideal of S[x1,x2,,xn].

Let C be a non-empty subsemigroup of a commutative semigroup S. Let ρ be the relation defined on S×C by (x,a)ρ(y,b) if and only if cay=cxy for some c∈ C. Clearly ρ is a congrurence on S×C. Then C1S={sc:sS,cC} is the quoitent semigroup of (S×C) modulo {ρ}, called the semigroup of fractions.

The composition on C1S is defined as (xa)(yb)=xyab. Also if I is an ideal of S, then C1I={ic:iI,cC} is an ideal of C1S, moreover C1I=C1I.

Theorem 2.33. Let C be a subsemigroup of a semigroup S and I be an ideal of S such that CI=ϕ. If I is a 2-absorbing primary ideal of S then C1I is a 2-absorbing primary ideal of C1S.

proof. Let I be a 2-absorbing primary ideal of S and (as)(br)(ct)C1I for some a,b,c ∈ S and s,r,t ∈ C. Hence there exists some p ∈ C such that abcp ∈ I. Therefore either ab ∈ I or bcpI or acpI. Now ab ∈ I implies (as)(br)=absrC1I, bcpI implies (br)(ct)=bcprtpC1I=C1I and acpI implies (as)(ct)=acpstpC1I=C1I, consequently, C1I is a 2-absorbing primary ideal of C1S.

Lemma 2.34. Let f:SS be a homomorphism of semigroups. Then the following statements holds

(1) f1( I )=f1( I ), where I' is an ideal of S'.

(2) If f is an isomorphism, then f(I)=f(I), where I is an ideal of S.

Theorem 2.35. Let f:SS be a homomorphism of semigroups. Then the following statements holds:

(1) If I' is a 2-absorbing primary ideal of S', then f1(I) is a 2-absorbing primary ideal of S.

(2) Let I be a proper ideal of S such that {(x,y)kerf:x=y}I×I. Then

(i) If f(I) is a 2-absorbing primary ideal of S', then I is a 2-absorbing primary ideal of S.

(ii) If f is onto and I is a 2-absorbing primary ideal of S, then f(I) is a 2-absorbing primary ideal of S'.

(3) If f is an isomorphism and I is a 2-absorbing primary ideal of S, then f(I) is a 2-absorbing primary ideal of S'.

Proof. (1) The proof is similar as that of ([3],Theorem 2.20(1)).

(2) The proof of (i) and (ii) are similar as that of Theorem 2.18 of [4] by replacing 2-absorbing ideal I as 2-absorbing primary ideal of S and then use the result of (1).

(3) It is trivial.

As a simple consequence of above theorem, we have the following result

Corollary 2.36. (1) Let SS be an extension of semigroup S and I be a 2-absorbing primary ideal of S'. Then IS is a 2-absorbing primary ideal of S. (2) Let IJ be two ideals of S. Then J is a 2-absorbing primary ideal of S if and only if J/ I is a 2-absorbing primary ideal of S/ I.

Lemma 2.37. Let S=S1×S2, where each Si is a semigroup. Then the following statement holds

(1) If I1 is an ideal of S1, then I1×S2=I1×S2.

(2) If I2 is an ideal of S2, then S1×I2=S1×I2. (3) I1×I2=I1×I2.

Theorem 2.38. Let S=S1×S2, where each Si is a semigroup. Then the following statements holds

(1) I1 is a 2-absorbing primary ideal of S1 if and only if I1×S2 is a 2-absorbing primary ideal of S.

(2) I2 is a 2-absorbing primary ideal of S2 if and only if S1×I2 is a 2-absorbing primary ideal of S.

Proof. (1) Suppose I1×S2 is a 2-absorbing primary ideals of S. Let abcI1 for some a,b,cS1. Then (abc,x3)I1×S2 for some xS2. So (a,x)(b,x)(c,x)I1×S2 and hence either (a,x)(b,x)I1×S2 or (b,x)(c,x)I1×S2 or (a,x)(c,x)I1×S2 and so either abI1 or bcI1 or acI1, which implies I1 is a 2-absorbing primary ideal of S1.

Conversely, Suppose I1 is a 2-absorbing primary ideal of S1. Let (a,x)(b,x)(c,x)I1×S2 for some a,b,c ∈ I1 and x ∈ S. Then abcI1 implies ab ∈ I1 or bcI1 or acI1 and hence either (a,x)(b,x)I1×S2 or (b,x)(c,x)I1×S2 or (a,x)(c,x)I1×S2, which implies I1×S2 is a 2-absorbing primary ideal of S1×S2.

(2) The proof is similar to (1).

Theorem 2.39. Let S=S1×S2, where each Si is a semigroup. Let I1 and I2 are ideals of S1 and S2 respectively. If I=I1×I2 is a 2-absorbing primary ideal of S then I1 and I2 are 2-absorbing primary ideals of S1 and S2 respectively.

Proof. Let abcI1 for some a,b,c∈ S. Then (a,x)(b,x)(c,x)=(abc,x3)I1×I2 for some x ∈ I2. Since I1×I2 is a 2-absorbing primary ideal of S, either (a,x)(b,x)=(ab,x2)I1×I2 or (b,x)(c,x)=(bc,x2)I1×I2=I1×I2 or (a,x)(c,x)=(ac,x2)I1×I2=I1×I2 (cf. Lemma 2.37(3)). Hence abI1 or bcI1 or acI1. Consequently I1 is a 2-absorbing primary ideal of S1. Similarly, we can prove that I2 is a 2-absorbing primary ideal of S2.

Remark 2.40. The following example shows that converse of Theorem 2.39 is not true. Consider the 2-absorbing primary ideals I1=6 and I2=5 of the semigroup (,.). Then (2,3)(3,2)(5,5)I1×I2 but nether (2,3)(3,2)I1×I2 nor (2,3)(5,5)I1×I2 nor (3,2)(5,5)I1×I2. Hence I1×I2 is not a 2-absorbing primary ideal of ×.

Theorem 2.41. Let S=S1×S2, where S1, S2 are commutative semigroup with zero and identity. Let I be a proper ideal of S. Then the following statements are equivalent

(1) I is a 2-absorbing primary ideal of S.

(2) Either I=I1×S2 for some 2-absorbing primary ideal I1 of S1 or I=S1×I2 for some 2-absorbing primary ideal I2 of S2 or I=I1×I2 for some primary ideals I1 of S1 and I2 of S2 respectively.

Proof. The proof is similar to ([14], Theorem 2.17).

### Acknowledgements.

The authors express their deep gratitude to learned referees for their meticulous reading and valuable suggestions which have definitely improved the paper. The research work reported here is supported by UGC-JRF NET Fellowship (Award No. 11-04-2016-421922) to Biswaranjan Khanra by the University Grants Commission, Governtment of India.

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