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### Article

Kyungpook Mathematical Journal 2021; 61(4): 711-725

Published online December 31, 2021

### 2-absorbing δ-semiprimary Ideals of Commutative Rings

Ece Yetkin Çelikel

Department of Electrical-Electronics Engineering, Faculty of Engineering, Hasan Kalyoncu University, Gaziantep, Turkey
e-mail : ece.celikel@hku.edu.tr, yetkinece@gmail.com

Received: December 18, 2020; Accepted: May 18, 2021

Let R be a commutative ring with nonzero identity, I(R) the set of all ideals of R and δ:I(R)I(R) an expansion of ideals of R. In this paper, we introduce the concept of 2-absorbing δ-semiprimary ideals in commutative rings which is an extension of 2-absorbing ideals. A proper ideal I of R is called 2-absorbing δ-semiprimary ideal if whenever a,b,cR and abcI, then either abδ(I) or bcδ(I) or acδ(I). Many properties and characterizations of 2-absorbing δ-semiprimary ideals are obtained. Furthermore, 2-absorbing δ1-semiprimary avoidance theorem is proved.

Keywords: 2-absorbing δ-primary ideal, δ-primary ideal, δ-semiprimary ideal, 2-absorbing ideal, 2-absorbing primay ideal

In this paper, all rings are commutative with nonzero identity. Let I be a proper ideal of a ring R and let I(R) denote the set of all ideals of R. The radical of I is defined by {aR:anIforsomen}, denoted by I. Let J be an ideal of R. Then the ideal (I:J) consists of r∈ R with rJI, that is, (I:J)={rR:rJI}. For undefined notations and terminology refer to [10].

Various generalizations of prime and primary ideals are studied extensively in [1]-[3],[13],[14]. Recall from [4] and [5] that a proper ideal I of R is called a (weakly) 2-absorbing ideal if whenever a,b,cR and (0abcI) abcI,then either abIor acI or bcI. A proper ideal I of R is called a (weakly) 2-absorbing primary ideal as in [6] and [7] if whenever a,b,cR and (0abcI) abcI,then either abIor acI or bcI. As in the recent study [13], a proper ideal I of R is said to be a 2-absorbing quasi-primary if I is a 2-absorbing ideal; or equivalently, if whenever a,b,cR and abcI,then either abIor acI or bcI. D. Zhao [14] introduced the concept of expansions of ideals and extended many results of prime and primary ideals into the new concept. From [14], a function δ from I(R) to I(R) is an ideal expansion if it has the following properties: Iδ(I) and if IJ for some ideals I,J of R, then δ(I)δ(J). For example, δ0 is the identity function where δ0(I)=I for all ideals I of R, and δ1 is defined by δ1(I)=I. For other examples, consider the functions δ+ and δ of I(R) defined by δ+(I)=I+J and δ(I)=(I:J) for all II(R), where J is an ideal of R, respectively (see [3]). He called a δ-primary ideal I of R if ab∈ I and aI for some a,bR imply bδ(I). Recently, (weakly) δ-semiprimary ideals are studied in [8]. A proper ideal I of R is called a (weakly) δ-semiprimary if whenever a,bR and (0abI) abI, then aδ(I) or bδ(I).

In this paper, we introduce and study 2-absorbing δ-semiprimary and weakly 2-absorbing δ-semiprimary ideals of commutative rings. Let δ:I(R)I(R) be an expansion of ideals of a ring R. We call a proper ideal I of R a (weakly) 2-absorbing δ-semiprimary ideal if whenever a,b,cR and (0abcI) abcI, then either abδ(I) or bcδ(I) oracδ(I). From the definitions, we have the following implications: 2-absorbing δ-primary (1) 2-absorbing δ-semiprimary (2) weakly 2-absorbing δ-semiprimary. Among many results in this study, it is shown in Example 2.3. and Example 2.4. that the implications (1) and (2) are not reversible. Also it shown in Theorem 2.17. that if I0, then the implication (2) is reversible. It is shown (Theorem 2.24.) that if δ(0) a 2-absorbing δ-semiprimary ideal with δ(δ(0))=δ(0) and I is a weakly 2-absorbing δ-semiprimary ideal, then either I is a 2-absorbing δ-semiprimary ideal of R or I2 is a 2-absorbing δ-semiprimary ideal of R. It is shown (Theorem 2.8.) that if II1I2In where I1,I2,...,Inn2) are ideals of R such that at most two of them are not 2-absorbing δ1-semiprimary, then IIi for some 1in. From Theorem 2.30. to Theorem 2.34., we characterize 2-absorbing δ-semiprimary ideals and weakly 2-absorbing δ-semiprimary ideals of R=R1××Rn where n2. Moreover, we state and prove 2-absorbing δ1-semiprimary avoidance theorem (Theorem 2.9.). In Section 3, we search an answer to the question that if I is a weakly 2-absorbing δ-semiprimary ideal of R and if 0JKLI for some ideals J, K, L of R, then does it imply either JKδ(I) or KLδ(I) or JLδ(I) ?

### 2. Properties of 2-absorbing δ-semiprimary Ideals

Definition 2.1. Let δ:I(R)I(R) be an expansion of ideals of R and I a proper ideal of R.

• 1. We call I a 2-absorbing δ-semiprimary ideal if whenever a,b,cR and abcI, then either abδ(I) or bcδ(I) or acδ(I).

• 2. We call I a weakly 2-absorbing δ-semiprimary ideal if whenever a,b,cR and 0abcI, then either abδ(I) or bcδ(I) or acδ(I).

Theorem 2.2. Let I be a proper ideal of R. Then the following statements hold:

• 1. If I is a 2-absorbing δ-semiprimary ideal, then I is a weakly 2-absorbing δ-semiprimary ideal.

• 2. I is a (weakly) 2-absorbing δ0-semiprimary ideal if and only if I is a (weakly) 2-absorbing ideal.

• 3. I is a (weakly) 2-absorbing δ1-semiprimary ideal if and only if I is a (weakly) 2-absorbing quasi primary ideal.

• 4. If I is a (weakly) delta-semiprimary idea, then I is a (weakly) 2-absorbing δ-semiprimary ideal.

• 5. If I is a (weakly) 2-absorbing δ-primary ideal, then I is a (weakly) 2-absorbing δ-semiprimary ideal.

• 6. Let δ and γ be two ideal expansions with δ(I)γ(I). If I is a (weakly) 2-absorbing δ-semiprimary ideal of R, then I is a (weakly) 2-absorbing γ-semiprimary ideal of R.

The following example shows that the concepts of weakly 2-absorbing δ-semiprimary and 2-absorbing δ-semiprimary are different.

Example 2.3. Let R=210 and δ:I(R)I(R) an expansion of ideals of R defined by δ(I)=(I:J) where J=7R. Then I=0 is a weakly 2-absorbing δ-semiprimary ideal of R by definition. Observe that δ(I)=30R. However, I is not 2-absorbing δ-semiprimary since 3514I but neither 35δ(I) nor 314δ(I) nor 514δ(I).

For a 2-absorbing δ-semiprimary ideal which is not 2-absorbing δ-primary, see the next example:

Example 2.4.([6, Example 2.9]) Let R=Z[X,Y,Z] and consider anideal I=(XYZ,X3Y3)R of R. Then I is a 2-absorbing δ1-semiprimary ideal of R but I is not a 2-absorbing δ1-primary ideal of R since XYZ∈ I but neither XY∈ I nor YZδ1(I) nor XZδ1(I) where δ1(I)=(XY)R.

Theorem 2.5. Let δ be an expansion function of I(R) and I a proper ideal of R.

• 1. If δ(I) is a (weakly) 2-absorbing ideal of R, then I is a (weakly) 2-absorbing δ-semiprimary ideal of R.

• 2. Let δ(δ(I))=δ(I). Then δ(I) is a (weakly) 2-absorbing δ-semiprimary ideal of R if and only if δ(I) is a (weakly) 2-absorbing ideal of R. Moreover, if δ(I) is 2-absorbing δ-semiprimary, then Min(δ(I))2.

Proof. (1) Suppose that (0abcI) abcI and abδ(I). Since Iδ(I) and δ(I) is 2-absorbing, we have acδ(I) or bc∈δ(I). Thus I is a (weakly) 2-absorbing δ-semiprimary ideal.

(2) Suppose that δ(I) is a (weakly) 2-absorbing δ-semiprimary ideal of R, since δ(δ(I))=δ(I),δ(I) is (weakly) 2-absorbing by the definition. The converse part is clear from (1). Suppose that δ(I) is a 2-absorbing δ-semiprimary ideal of R. Then δ(I) is 2-absorbing; and so, Min(δ(I))2 by [4, Theorem 2.3.].

The converse of Theorem 2.5. (1) is also true for δ=δ1 by [13, Proposition 2.5].

Theorem 2.6. Let δ be an expansion of I(R) and I a 2-absorbing δ-semiprimary ideal of R. If δ(I)δ(I), then I is a 2-absorbing δ-semiprimary ideal of R. In particular, if δ=δ1, then I is a 2-absorbing ideal of R.

Proof. Let a,b,cR with abcI and abδ(I). Then anbncnI for some positive integer n ≥ 1. Since δ(I)δ(I), we conclude anbnδ(I). Since I is 2-absorbing δ-semiprimary, we have bncnδ(I) or ancnδ(I). Since δ(I)δ(I), we conclude that bcδ(I) or acδ(I), we are done. The particular case is clear from [13, Proposition 2.5].

The next example shows that the converse of Theorem 2.6. is not satisfied in general.

Example 2.7. Consider the ideal I=(X3)/(X4) of R=12[X]/(X4). Then I=(6,X)/(X4) is a 2-absorbing ideal (2-absorbing δ0-semiprimary ideal). However, I is not a 2-absorbing δ0-semiprimary ideal since 0(X+(X4))(X+(X4))(X+(X4))I but X2+(X4)δ0(I)=I.

Theorem 2.8. Let I1,I2,...,In(n2) be ideals of R such that at most two of them are not 2-absorbing δ1-semiprimary. If II1I2In, then IIi for some 1in.

Proof. From the covering II1I2In, we conclude that II1I2In. By our hypothesis, we may assume that Ik is a 2-absorbing δ1-semiprimary ideal for all k3. Hence Ik is 2-absorbing for all k3 by Theorem 2.6. Thus I=IIi=Ii for some 1in by [12, Theorem 3.1].

Let I,I1,I2,...,In be ideals of R. Recall that an efficient covering of R is a covering II1I2In in which no Ij where 1jn satisfies IIj (i.e., no Ij is superfluous.) In the following result, we obtain 2-absorbing δ1-semiprimary avoidance theorem.

Theorem 2.9. (2-absorbing δ1-semiprimary avoidance theorem) Let I1, I2,...,In(n2) be ideals of R such that at most two of them not 2-absorbing δ1-semiprimary. Suppose that Ii(Ij:x) for all xR\Ij for all ij. If II1I2In, then IIi for some 1in.

Proof. Assume on the contrary that IIi for all 1in. Hence II1I2In is an efficient covering of ideals of R. So, it is clear that I=(II1)(II2)(IIn) is an efficient union. From [11, Lemma 2.1], ik(IIi)IIk. Since at most two of the Ii are not 2-absorbing δ1-semiprimary ideals, we may assume that Ik is a 2-absorbing δ1-semiprimary ideal. From Theorem 2.6., we conclude that Ik is 2-absorbing; and so (Ik:x) is a prime ideal of R for all xIi by [4, Theorem 2.5.]. On the other hand, our hypotesis implies that there exist aiIi\(Ik:x) for all i≠ k. Hence there are positive integers mi such that aimiIi. Put a=ikai and m=max{m1,m2,...,mn}. It is clear that amxik(IIi). We show that amxIIk. Assume that amxIIk. Hence amxIk, and so am(Ik:x). Since (Ik:x) is prime, we conclude that ai(Ik:x) for some ik, a contradiction. Thus ik(IIi)\(IIk) is nonempty which contradicts with ik(IIi)IIk Thus IIi for some 1in.

Recall from [14] that an ideal expansion δ of I(R) is said to be intersection preserving if it satisfies δ(I1I2In)=δ(I1)δ(I2)δ(In) for any ideals I1,I2,...,Inof R.

Theorem 2.10. Let δ be an intersection preserving expansion function of I(R). If I1,I2,...,In are 2-absorbing δ-semiprimary ideals of R with δ(Ii)=K for all i{1,2,...,n}, then I= i=1nIi is a 2-absorbing δ-semiprimary ideal of R.

Proof. Suppose that abcI, abδ(I) and bcδ(I) for some a,b,cR. Since δ(I)=δ( i=1nIi)= i=1nδ(Ii)=K, we have abK and bcK. Since abcIi and Ii is 2-absorbing δ-semiprimary, acδ(Ii)=K=δ(I); so we are done.

Theorem 2.11. Let δ be an expansion function of I(R) and I a (weakly) 2-absorbing δ-semiprimary ideal of R. Then the following hold:

• 1. If JI and δ(J)=δ(I), then J is a (weakly) 2-absorbing δ-semiprimary ideal of R.

• 2. If JI0, then J is a (weakly) 2-absorbing quasi-primary ideal of R.

• 3. Let I0. If K is an ideal of R, then IK, IK and In are (weakly) 2-absorbing quasi-primary ideals of R for all positive integers n1

Proof. (1) Suppose that (0abcJ)abcJ for some a,b,cR. Since JI, we have (0abcI) abcI. Since I is (weakly) 2-absorbing δ-semiprimary, we have either abδ(I) or bcδ(I) or acδ(I). Since δ(I)=δ(J), J is a (weakly) 2-absorbing δ-semiprimary ideal of R.

(2) Since JI0, we have δ1(J)=J=I=δ1(I)=0. Thus the result is clear from (1).

(3) is a particular result of (2).

Theorem 2.12. Let δ be an expansion of I(R). Every proper principal ideal is a 2-absorbing δ-semiprimary ideal of R if and only if every proper ideal is a 2-absorbing δ-semiprimary ideal of R.

Proof. Suppose that every proper principal ideal is a 2-absorbing δ-semiprimary ideal of R. Let I be a proper ideal of R and a,b,c∈ R with abc ∈ I. Then abc∈(abc) and since (abc) is 2-absorbing δ-semiprimary ideal of R by our assumption, we have either abδ(abc)δ(I) or bcδ(abc)δ(I) or acδ(abc)δ(I) or Thus I is a 2-absorbing δ-semiprimary ideal of R. The converse part is obvious.

Theorem 2.13. Let δ be an ideal expansion such that δ(I)I and δ(I) a semiprime ideal of R for every ideal I. If I is a 2-absorbing δ-semiprimary ideal, then δ(I)=I.

Proof. Suppose that aI. Then there exists n which is the least positive integer n with anI. If n=1, then aIδ(I). For n2, an=an2aaI. Since an1I, we have a2δ(I). Since δ(I) is semiprime, aδ(I). Thus δ(I)=I.

Recall that a ring R is said to be a Boolean ring if x2=x for every x∈ R. Since I=I for every proper ideal I of R, we have the following result.

Theorem 2.14. Let R be a Boolean ring and I a proper ideal of R. Then the following are equivalent:

• 1. I is a (weakly) 2-absorbing quasi-primary ideal of R.

• 2. I is a (weakly) 2-absorbing primary ideal of R.

• 3. I is a (weakly) 2-absorbing ideal of R.

Proof. Since I=I, the claim is clear.

Definition 2.15. Let δ be an expansion function of I(R) and I a weakly 2-absorbing δ-semiprimary ideal of R. We call a triple (a,b,c) a δ-triple zero of I if abc=0 for some elements a,b,c of R (not necessarily distinct) and neither abδ(I) nor bcδ(I) nor acδ(I).

Remark 2.16. Let δ be an expansion function of I(R) and I a weakly 2-absorbing δ-semiprimary ideal of R. Then I is a not 2-absorbing δ-semiprimary ideal of R if and only if there exists at least one δ-triple zero of I.

Theorem 2.17. Let δ be an expansion function of I(R) and I a weakly 2-absorbing δ-semiprimary ideal of R. If I is not a 2-absorbing δ-semiprimary ideal of R, then I3=0; that is I0.

Proof. Suppose that I is a weakly 2-absorbing δ-semiprimary ideal of R which is not 2-absorbing δ-semiprimary. Hence there exists a δ-triple zero (a,b,c) of I by Remark 2.16. Then abc=0 and neither abδ(I) nor bcδ(I) nor acδ(I). for some b,cR. First, we show that abI=0. Assume that abi≠0 for some i∈ I. Since 0ab(c+i)I and I is a weakly 2-absorbing δ-semiprimary, we conclude abI or a(c+i)I or b(c+i)I, a contradiction. Similarly, it is easy to show that bcI=acI=0. Now, we show that aI2=0. Assume that ai1i20 for some i1,i2I. Since abI=bcI=acI=0, we have 0a(b+i1)(c+i2)=ai1i2I. Since I is a weakly 2-absorbing δ-semiprimary, this contradicts our assumption that abδ(I) nor bcδ(I) nor acδ(I). Thus aI2=0. One can easily show symmetrically that bI2=cI2=0. Lastly, we show that I3=0. Assume that i1i2i30 for some i1,i2,i3I. Since abI=bcI=acI=aI2=bI2=cI2=0, observe that 0(a+i1)(b+i2)(c+i2)=i1i2i3I. Since I is a weakly 2-absorbing δ-semiprimary, again we conclude a contradiction. Thus I3=0.

The following example shows that for an ideal I of R with I3=0, I needs not to be a weakly 2-absorbing δ-semiprimary ideal.

Example 2.18. Let R=60 and I=30R. Then I3=0. However, I is not a weakly 2-absorbing δ-semiprimary ideal (for δ=δ0 or δ=δ1) since 235I but neither 23δ(I) nor 35δ(I) nor 25δ(I).

As a conclusion of Theorem 2.17., we have the following two results.

Corollary 2.19. Let R be a reduced ring. Then every nonzero weakly 2-absorbing δ-semiprimary ideal of R is a 2-absorbing δ-semiprimary ideal of R.

Corollary 2.20. Let M be a finitely generated R-module. Let I be a weakly 2-absorbing δ-semiprimary ideal of R that is not 2-absorbing δ-semiprimary. If IM=M, then M=0.

Theorem 2.21 Let δ be an expansion function of I(R) and I a weakly 2-absorbing δ-semiprimary ideal of R with δ(I)=δ(0). Then I is not 2-absorbing δ-semiprimary ideal of R if and only if there exists a δ-triple zero of 0.

Proof. Suppose that I is not a 2-absorbing δ-semiprimary ideal. Hence abc=0 but abδ(I), acδ(I) and bcδ(I) for some a,b,cR. Since δ(I)=δ(0),(a,b,c) is a δ-triple zero of 0. The converse part is obvious.

In particular, suppose that I0 is a weakly 2-absorbing quasi primary ideal of R. A a consequence of Theorem 2.21., there exists a δ-triple zero of 0 if and only if I is not a 2-absorbing quasi primary ideal of R. The following example shows that if δ(I)δ(0) and there exists a δ-triple zero of 0, then I may be a 2-absorbing δ-semiprimary ideal of R.

Example 2.22. Consider R=24, δ:I(R)I(R) is defined by δ(I)=δ1(I) for every nonzero proper ideal I of R, and δ(0)=δ0(0)=0. Let I=12R. Then δ(I)=6R is 2-absorbing by [4, Theorem 3.15], I is a 2-absorbing δ-semiprimary ideal by Theorem 2.5. Since 234=0 but neither 23δ(0) nor 24δ(0) nor 34δ(0), we conclude that (2,3,4) is δ-triple zero of 0.

Theorem 2.23. δ be an intersection preserving expansion function of I(R). If I1,I2,...,In are weakly 2-absorbing quasi-primary (weakly 2-absorbing δ1-semiprimary) ideals of R that are not 2-absorbing quasi-primary, then I=i=1nIi is a weakly 2-absorbing quasi-primary ideal of R.

Proof. Since each Ii is a 2-absorbing quasi-primary ideal of R that is not 2-absorbing quasi-primary, we have δ(Ii)=Ii=0 by Theorem 2.17. Thus, remain of the proof is easily concluded similar to the proof of Theorem 2.10.

Theorem 2.24. Let δ be an expansion function of I(R) and δ(0) a 2-absorbing δ-semiprimary ideal such that δ(δ(0))=δ(0). Suppose that I is a weakly 2-absorbing δ-semiprimary ideal. Then I is a 2-absorbing δ-semiprimary ideal of R or I2 is a 2-absorbing δ-semiprimary ideal of R.

Proof. Suppose that I is a weakly 2-absorbing δ-semiprimary ideal that is not 2-absorbing δ-semiprimary. Hence I30δ(0) by Theorem 2.17. Since δ(0) is a 2-absorbing ideal of R by Theorem , we conclude that I2δ(0) by [4, Theorem 2.13.]. Since 0I2δ(0) and δ(δ(0))=δ(0), we have δ(I2)=δ(0). Since δ(I2) is a 2-absorbing ideal of R,I2 is a 2-absorbing δ-semiprimary ideal of R by Theorem 2.5.

Let R and S be commutative rings with 1 ≠ 0, and let δ,γ be two expansion functions of I(R) and I(S), respectively. Then a ring homomorphism f:RS is called a δγ-homomorphism if δ(f1(I))=f1(γ(I))for all ideals I of S For example, if γ1is a radical operation on ideals of S and δ1 is a radical operation on ideals of R. Then every homomorphism from R to S is a δ1γ1-homomorphism. Additionally, if f is a δγ-epimorphism and I is an ideal of R containing ker(f), then γ(f(I))=f(δ(I)) [3]

Theorem 2.25. Let f:RS be a δγ-homomorphism, where δ and γ are expansion functions of I(R) and I(S), respectively. Then the following statements hold:

• 1. If J is a 2-absorbing γ-semiprimary ideal of S, then f1J is a 2-absorbing δ-semiprimary ideal of R.

• 2. If J is a weakly 2-absorbing γ-semiprimary ideal of S, and ker(f) is a weakly 2-absorbing δ-semiprimary ideal of R, then f1J is a weakly 2-absorbing δ-semiprimary ideal of R.

• 3. Let f be an epimorphism and I a proper ideal of R with ker(f)I. Then I is (weakly) 2-absorbing δ-semiprimary ideal of R if and only if fI is a (weakly) 2-absorbing γ-semiprimary ideal of S.

Proof. (1) Let abcf1(J) for some a,b,cR. Then f(abc)=f(a)f(b)f(c), which implies f(a)f(b)=f(ab)γ(J) or f(b)f(c)=f(bc)γ(J) or f(a)f(c)=f(bc)γ(J). Thus we have abf1(γ(J)) or bcf1(γ(J)) or acf1(γ(J)). Since f1(γ(J))=δ(f1(J)),f1(J) is a 2-absorbing δ-semiprimary ideal of R.

(2) Let 0abcf1(J) for some a,b,cR. Then f(abc)=f(a)f(b)f(c)J. If f(abc)0, it can be easily proved similar to (1) that f1(J) is a weakly 2-absorbing δ-semiprimary ideal of R. Assume that f(abc)=0. Hence abcker(f). Since ker(f) is weakly 2-absorbing δ-semiprimary, we have abδ(ker(f)) or bcδ(ker(f)) or acδ(ker(f)). Sinceδ(ker(f))=δ(f1(0))δ(f1(J)), we are done.

(3) Let (0xyzf(I)) xyzf(I) for some x,y,zS. Then there are some elements a,b,cI such that x=f(a), y=f(b) and z=f(c). Then f(a)f(b)f(c)=f(abc)f(I) and since ker(f)I, we conclude (0abcI) abc ∈ I. Since I is (weakly) 2-absorbing δ-semiprimary, we have either abδ(I) or bcδ(I) or acδ(I). Thus xyf(δ(I)) or yzf(δ(I)) or yzf(δ(I)). Since f(δ(I))=δ(f(I)), we are done.

Remark 2.26. Let δ be an expansion function of I(R) and I a proper ideal of R. Then the function δq:R/IR/I defined by δq(J/I)=δ(J)/I for all ideals IJ,becomes an expansion function of R/I [3]. Consider the natural homomorphism π:RR/J. Then for ideals I of R with ker(π)I, we have δq(π(I))=π(δ(I)).

From Theorem 2.25. and Remark 2.26., we have the following result.

Corollary 2.27. Let δ be an expansion function of I(R).

• 1. Let I and J be ideals of R with IJ. Then J is a 2-absorbing δ-semiprimary ideal of R if and only if J/I is a 2-absorbing δq-semiprimary ideal of R/I.

• 2. If I is a 2-absorbing δ-semiprimary ideal of R and R is a subring with RI, then IR is a 2-absorbing δ-semiprimary ideal of R.

Let δ be an expansion function of ideals of a polynomial ring R[X] where X is an indeterminate. Observe that the function as in Remark 2.26., δq:R[X]/(X)R[X]/(X) defined by δq(J/(X))=δ(J)/(X) for all ideals J of R[X] with (X)⊆ J, is an expansion function of ideals of R as R[X]/(X)R. According to these expansions, we have the following equivalent situations:

Theorem 2.28. Let δ be an expansion function of I(R) and I a proper ideal of R. Then the following are equivalent:

• 1. I is a 2-absorbing δq-semiprimary ideal of R.

• 2. (I,X) is a 2-absorbing δ-semiprimary ideal of R[X].

Proof. From Corollary 2.27., we conclude that (I,X) is a 2-absorbing δ-semiprimary ideal of R[X] if and only if (I,X)/(X) is a 2-absorbing δq-semiprimary ideal of R[X]/(X). Since (I,X)/(X)I and R[X]/(X)R, the result is obtained.

Let S be a multiplicatively closed subset of a ring R and let δ be an expansion function of I(R). Note that δS is an expansion function of I(RS) such that δS(IS)=(δ(I))S. In the next theorem, we investigate 2-absorbing δS-semiprimary ideals of the localization RS.

Theorem 2.29. Let δ be an expansion function of I(R) and S a multiplicatively closed subset of R. If I is a (weakly) 2-absorbing δ-semiprimary ideal of R with IS=, then IS is a (weakly) 2-absorbing δS-semiprimary ideal of RS.

Proof. Let (0xs1ys2zs3IS) xs1ys2zs3IS for some x,y,zR; s1,s2,s3S. Then we have (0sxyzI) sxyz∈ I for some s∈ S. Then sxyδ(I) or yzδ(I) or sxzδ(I). Hence sxyss1s2δ(I)S or yzs2s3δ(I)S or sxzss1s3δ(I)S. Since (δ(I))S=δS(IS), IS is a (weakly) 2-absorbing δS-semiprimary ideal of RS.

Let R=R1××Rn(n2) where R1,R2,...,Rn are commutative rings with nonzero identity, let δi be an expansion function of I(Ri) for each i{1,2,...,n}. For a proper ideal I1××In, the function δ× defined by δ×(I1×I2×...×In)=δ1(I1)×δ2(I2)×...×δn(In) is an expansion function of I(R). In the next four theorems, we characterize 2-absorbing δ-semiprimary ideals and weakly 2-absorbing δ-semiprimary ideals of R1××Rn.

Theorem 2.30. Let R1 and R2 be commutative rings with 1≠0 and R=R1×R2, and let δ1, δ2 be expansion functions of mathcalI(R1) and I(R2), respectively. Suppose that δ×(I) is a proper ideal of R for any proper ideal of R. Then the following statements are equivalent:

• 1. I=I1×I2 is a 2-absorbing δ×-semiprimary ideal of R.

• 2. Either I1 is a 2-absorbing δ1-semiprimary ideal ofR1 and δ2(I2)=R2 or I2 is a 2-absorbing δ2-semiprimary ideal of R2 and δ1(I1)=R1 or I1,I2 are δ1,2-semiprimary ideals of R1,R2, respectively.

Proof. (1)(2): Suppose that I=I1×I2 is a 2-absorbing δ×-semiprimary ideal of R. Since I is proper, δ×(I)=δ1(I1)×δ2(I2) is a proper ideal of R from the hypothesis. Hence we have three cases:

Case 1: Let δ1(I1)R1 and δ2(I2)=R2. We show that I1 is a 2-absorbing δ1-semiprimary ideal of R1. Suppose that abcI1 and abδ1(I1).Then (a,0)(b,0)(c,0)I and (a,0)(b,0)δ×(I) implies that (b,0)(c,0)δ×(I) or (a,0)(c,0)δ×(I). Thus bcδ1(I1) or acδ1(I1), we are done.

Case 2: Let δ2(I2)R2 and δ1(I1)=R1. One can easily obtain similar to Case 1 that I2 is a 2-absorbing δ2-semiprimary ideal of R2.

Case 3: Let δ1(I1)R1 and δ2(I2)R2. Suppose that abI1 and aδ1(I1) for some a,bR1. Observe that (a,1)(b,1)(1,0)I, (a,1)(b,1)δ×(I), and (a,1)(1,0)δ×(I). Since I is 2-absorbing δ×-semiprimary, we conclude (b,1)(1,0)δ×(I). Thus bδ1(I1); and so I1 is δ1-semiprimary ideal of R1. It can be shown by a symmetric way that I2 is a 2-absorbing δ2-semiprimary ideal of R2.

(2)(1): If I1 is a 2-absorbing δ1-semiprimary ideal of R1 and δ2(I2)=R2 or I2 is a 2-absorbing δ2-semiprimary ideal of R2 and δ1(I1)=R1, then clearly I is a 2-absorbing δ×-semiprimary ideal of R. Now, suppose that I1 and I2 are δ1,2-semiprimary ideals of R1,R2, respectively. Suppose that (a1,a2)(b1,b2)(c1,c2)I=I1×I2, (a1,a2)(b1,b2)δ×(I) and (a1,a2)(c1,c2)δ×(I). Here we have four cases.

Case 1: Let a1b1δ1(I1) and a1c1δ1(I1) .Since a1b1c1I1, it contradicts with the assumption that I1 is a δ1-semiprimary ideal.

Case 2: Let a2b2δ2(I2) and a2c2δ2(I2). Since a2b2c2I2, it contradicts with the assumption that I2 is a δ2-semiprimary ideal.

Case 3: Let a1b1δ1(I1) and a2c2δ2(I2). Since a1b1c1I1 and I1 is δ1-semiprimary, we have c1δ1(I1). Since a2b2c2I2 and I2 is δ2-semiprimary, we have b2δ2(I2). Thus (b1,b2)(c1,c2)δ×(I). textbf{Case 4: }Let a1c1δ1(I1) and a2b2δ2(I2). Since a1b1c1I1 and I1 is δ1-semiprimary, we have b1δ1(I1). Since a2b2c2I2 and I2 is δ2-semiprimary, we have c2δ2(I2). Thus (b1,b2)(c1,c2)δ×(I). Therefore, I is a 2-absorbing δ×-semiprimary ideal of R.

Theorem 2.31. Let R1,R2,...,Rn be commutative rings with nonzero identity and R=R1××Rn where n2. Let δi be an expansion function of I(Ri) for each i=1,...,n. Then the following statements are equivalent:

• 1. I is a 2-absorbing δ×-semiprimary ideal of R.

• 2. I=I1××In and either for some k{1,...,n} such that Ik is a 2-absorbing δ×-semiprimary ideal of Rk and δj(Ij)=Rj for all j{1,...,n}\{k} or Ik and It are δk,t-semiprimary ideals of Rk and Rt, respectively for some k,t{1,2,...,n} and δj(Ij)=Rj for all j{1,...,n}\{k,t}

Proof. It can be obtained from Theorem 2.30. by using mathematical induction on n.

Theorem 2.32. Let R1 and R2 be commutative rings with identity, R=R1×R2, and let δ1, δ2 be expansion functions of mathcalI(R1) and I(R2), respectively. Then the following statements are equivalent:

• 1. I=I1×R2 is a weakly 2-absorbing δ×-semiprimary ideal of R.

• 2. I=I1×R2 is a 2-absorbing δ×-semiprimary ideal of R.

• 3. I1 is a 2-absorbing δ1-semiprimary ideal of R1

Proof. (1)(2): Suppose that I=I1×R2 is a weakly 2-absorbing δ×-semiprimary ideal of R. Since I30, I=I1×R2 is a 2-absorbing δ×-semiprimary ideal of R by Theorem 2.17.

(2)(3)(1) is clear from Theorem 2.30.

Definition 2.33. Let R be a ring and δ an expansion function of I(R). We say δ has () property if the following condition is satisfied for all ideals J of R:

()δ(J)=Rif and only if J=R.

Theorem 2.34. Let R1,R2,...,Rn be commutative rings with identity and R=R1××Rn where n≥3. Let δi be an expansion function of I(Ri) which has () property for each i=1,...,n. For a nonzero ideal I of R, the following statements are equivalent:

1. I is a weakly 2-absorbing δ×-semiprimary ideal of R.

2. I is a 2-absorbing δ×-semiprimary ideal of R.

3. I=I1××In and either for some k{1,...,n} such that Ik is a 2-absorbing δ×-semiprimary ideal of Rk and Ij=Rj for all j{1,...,n}\{k} or Ik and It are δk,t-semiprimary ideals of Rk and Rt, respectively for some k,t{1,2,...,n} and Ij=Rj for all j{1,...,n}\{k,t}

Proof. (1)(2): Suppose that I=I1××In is a weakly 2-absorbing δ×-semiprimary ideal of R. Since I is nonzero, there exists an element 0(x1,x2,...,xn)I. Hence 0(x1,1,...,1)(1,x2,...,1)(1,1,...,xn)I implies that 1δ(Ik) for some k{1,...,n}. Thus Ik=Rk for some k{1,...,n}; so I3 can not be 0. Therefore, I is a 2-absorbing δ×-semiprimary ideal of R by Theorem 2.17. The converse is obvious.

(2)(3): From Theorem 2.31., the claim is clear.

Let R be a commutative ring and M an R-module. The idealization R(+)M={(r,m):rR,mM} is a commutative ring with addition and multiplication, respectively: (r,m)(s,m)=(r+s,m+m) and (r,m)(s,m)=(rs,rm+sm) for each r,sR;m,mM. Additionally, I(+)N is an ideal of R(+)M where I is an ideal of R and N is a submodule of M if and only if IMN ([2] and [9]). In this circumstances, I(+)N is called a homogeneous ideal of R(+)M. Let δ be an expansion function of R. Clear that δ(+) is defined as δ(+)(I(+)N)=δ(I)(+)M for all ideal I(+)N of R(+)M is an expansion function of R(+)M.

Theorem 2.35. Let δ be an expansion function of R and I(+)N be a homogeneous ideal of R(+)M. Then, I is a 2-absorbing δ-semiprimary ideal of R if and only if I(+)N is a 2-absorbing δ(+)-semiprimary ideal of R(+)M.

Proof. Let (r1,m1)(r2,m2)(r3,m3)=(r1r2r3,r2r3m1+r1r3m2+r1r2m3)I(+)N. Then r1r2r3I. Since I is 2-absorbing δ-semiprimary, we have r1r2δ(I) or r2r3δ(I) or r1r3δ(I) Since δ(+)(I(+)N)=δ(I)(+)M, we conclude that (r1,m1)(r2,m2)δ(+)(I(+)N) or (r2,m2)(r3,m3)δ(+)(I(+)N) or (r1,m1)(r3,m3)δ(+)(I(+)N). Conversely, suppose that r1r2r3I for some r1,r2,r3R. Then (r1,0),(r2,0)(r3,0)=(r1r2r3,0)I(+)N. The remain of the proof is clear.

### 3. Strongly (weakly) 2-absorbing δ-semiprimary ideal

First, we state the following theorem which gives a characterization for 2-absorbing δ-semiprimary ideals in terms of ideals of R.

Theorem 3.1. Let δ be an expansion function of I(R) and I a proper ideal of R. Then the following are equivalent:

• 1. I is a 2-absorbing δ-semiprimary ideal of R.

• 2. For every elements a,bR with abδ(I), (I:ab)(δ(I):a)(δ(I):b).

• 3. For every elements a,bR with abδ(I), (I:ab)(δ(I):a) or (I:ab)(δ(I):b).

• 4. For every elements a,bR with abJI and abδ(I) implies either aJδ(I) or bJδ(I).

• 5. For any ideals J,K and L of R with JKLI implies JKδ(I) or JLδ(I) or KLδ(I).

Proof. (1)(2): Let c(I:ab). Since abcI, abδ(I) and I is 2-absorbing δ-semiprimary, we have acδ(I) or bcδ(I). Hence c(δ(I):a)(δ(I):b).

(2)(3): It is straightforward.

(3)(4): Suppose that (3) holds and abJI and abδ(I). Hence, we have J(I:ab)(δ(I):a) or J(I:ab)(δ(I):b) by our assumption. Thus, aJδ(I) or bJδ(I).

(4)(5): Suppose that JKLI and KLδ(I). Then abδ(I) for some aK and bL. Hence aJδ(I) or bJδ(I). Assume that aJδ(I) and bJδ(I). We show that JKδ(I). If kJδ(I) for some kK, then (a+k)bJI. Since bJδ(I), we have (a+k)Jδ(I); and so, we get kJδ(I), a contradiction. Assume that aJδ(I) and bJδ(I). Similar to the previous argument, we conclude that JLδ(I). Now, suppose that aJδ(I) and bJδ(I). Assume that neither JKδ(I) nor JLδ(I). Then there exist kK and lL such that kJδ(I) and lJδ(I). Since klJI, we conclude klδ(I). Since (a+k)lJI, lJδ(I) and (a+k)J=aJ+kJδ(I), we have (a+k)lδ(I). Since klδ(I), we have alδ(I). Since (b+l)kJI, kJδ(I) and (b+l)J=bJ+lJδ(I), we have (b+l)kδ(I). Since klδ(I), we have bkδ(I). Since (a+k)(b+l)JI, (a+k)Jδ(I) and (b+l)Jδ(I), we have (a+k)(b+l)δ(I). Since al,bk,klδ(I), we conclude abδ(I), a contradiction. Thus Kδ(I) or JLδ(I).

(5)(1): Suppose that abcI for some a,b,cR. Put J=(a), K=(b) and L=(c) in (5). Hence, the result is clear.

Now, we define strongly (weakly) 2-absorbing δ-semiprimary ideals as follows:

Definition 3.2. Let δ be an expansion function of I(R). We call a proper ideal I of R a strongly (weakly) 2-absorbing δ-semiprimary ideal if whenever J,K,LI(R) with (0JKLI) JKLI implies JKδ(I) or JLδ(I) or KLδ(I).

As a result of Theorem 3.1., I is a 2-absorbing δ-semiprimary ideal of R if and only if I is a strongly 2-absorbing δ-semiprimary ideal of R. Motivated from this result, we search the answer for the following question:

Question 3.3. If I is a weakly 2-absorbing δ-semiprimary ideal of R, then does I need to be a strongly weakly 2-absorbing δ-semiprimary ideal of R?

Theorem 3.4. Let I be a weakly 2-absorbing δ-semiprimary ideal of R and suppose that 0JKLI for some ideals J,K and L of R. If I is a free δ-triple-zero with respect to JKL, then JKδ(I) or KLδ(I) or JLδ(I).

To prove the theorem above, we need the following definition and lemmas.

Definition 3.5. Let I be a weakly 2-absorbing δ-semiprimary ideal of R and suppose that JKLI for some ideals J,K and L of R. We call I a free δ-triple-zero with respect to JKL if (a,b,c) is not a δ-triple-zero of I for every a∈ J, b∈ K and c∈ L. (Equivalently, if a∈ J, b∈ K, c∈ L, then abδ(I) or bcδ(I) or acδ(I).)

Lemma 3.6. Let I be a weakly 2-absorbing δ-semiprimary ideal of R and let abLI for some a,b∈ R and an ideal L of R. If (a,b,l) is not a δ-triple-zero of I for all l∈ L and abδ(I), then aLδ(I) or bLδ(I).

Proof. Assume on the contrary that abLI but neither abδ(I) nor aLδ(I) nor bLδ(I). Hence there exist l1,l2L such that al1δ(I) and bl2δ(I). Since abl1I but neither abδ(I) nor al1δ(I), we have bl1δ(I) by our hypothesis that (a,b,l1) is not a δ-triple-zero of I. Similarly, since abl2I but neither abδ(I) nor bl2δ(I), we conclude that al2δ(I). Now ab(l1+l2)I and since abδ(I), we have either a(l1+l2)δ(I) or b(l1+l2)δ(I). Thus, we conclude al2δ(I) or bl2δ(I), a contradiction. Thus, aLδ(I) or bLδ(I).

Lemma 3.7. Let I be a weakly 2-absorbing δ-semiprimary ideal of R and let aKLI for some a∈ R and for an ideal J of R. If (a,k,l) is not a δ-triple-zero of I for all k∈ K, l∈ L, then aKδ(I) or aLδ(I) or KLδ(I).

Proof. Assume that neither aKδ(I) nor aLδ(I) nor KLδ(I). Thus there exist k,k1K such that akδ(I) and k1Lδ(I). Since akLI, akδ(I) and aLδ(I), we have kLδ(I) by Lemma 3.6. Since ak1LI, aLδ(I) and k1Lδ(I), we have by ak1δ(I) Lemma 3.6. Now, since a(k+k1)LI and aLδ(I), from Lemma 3.6. we conclude that either a(k+k1)δ(I) or (k+k1)LI. Hence akδ(I) or k1LI, a contradiction. Thus, aKδ(I) or aLδ(I) or KLδ(I).

Proof of Theorem 3.4. Assume on the contrary that neither JKδ(I) nor KLδ(I) nor JLδ(I). Hence there exists a,a1J such that aKδ(I) and a1Lδ(I). Since aKLI, KLδ(I) and aKδ(I), we have aLδ(I) by Lemma 3.7. Since a1KLI, KLδ(I) and a1Lδ(I), we have a1Kδ(I) by Lemma 3.7. Now (a+a1)KLI and since KLδ(I), we conclude either (a+a1)Kδ(I) or (a+a1)Lδ(I). Hence, we have aKδ(I) or a1Lδ(I), a contradiction. Therefore, JKδ(I) or KLδ(I) or JLδ(I).

More general than 2-absorbing δ-semiprimary ideal of a commutative ring, the concept of n-absorbing δ-semiprimary ideal where n is a positive integer can be defined. We shall give just the definition of this concept which may be inspired the other work:

Definition 3.8. Let R be a commutative ring with nonzero identity, δ:I(R)I(R) an expansion of ideals of R and n a positive integer. We call a proper ideal I of R a (weakly) n-absorbing δ-semiprimary ideal if whenever (0x1xn+1I)x1xn+1I for some x1,...,xn+1R, then there exists 1kn such that x1xk1xk+1xn+1δ(I). In particular, for n=1,2, it is δ-semiprimary and 2-absorbing δ-semiprimary ideal, respectively.

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