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 {a∈R:an∈I for some n∈ℕ}, denoted by I. Let J be an ideal of R. Then the ideal (I:J) consists of r∈ R with rJ⊆I, that is, (I:J)={r∈R:rJ⊆I}. 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,c∈R and (0≠abc∈I) abc∈I, then either ab∈I or ac∈I or bc∈I. A proper ideal I of R is called a (weakly) 2-absorbing primary ideal as in [6] and [7] if whenever a,b,c∈R and (0≠abc∈I) abc∈I, then either ab∈I or ac∈I or bc∈I. 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,c∈R and abc∈I, then either ab∈I or ac∈I or bc∈I. 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: I⊆δ(I) and if I⊆J 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 I∈I(R), where J is an ideal of R, respectively (see [3]). He called a δ-primary ideal I of R if ab∈ I and a∉I for some a,b∈R imply b∈δ(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,b∈R and (0≠ab∈I) ab∈I, then a∈δ(I) or b∈δ(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,c∈R and (0≠abc∈I) abc∈I, then either ab∈δ(I) or bc∈δ(I) orac∈δ(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 I⊈0, 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 I⊆I1∪I2∪⋯∪In where I1,I2,...,Inn≥2) are ideals of R such that at most two of them are not 2-absorbing δ1-semiprimary, then I⊆Ii for some 1≤i≤n. 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 n≥2. 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 0≠JKL⊆I for some ideals J, K, L of R, then does it imply either JK⊆δ(I) or KL⊆δ(I) or JL⊆δ(I) ?

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,c∈R and abc∈I, then either ab∈δ(I) or bc∈δ(I) or ac∈δ(I).

• 2. We call I a weakly 2-absorbing δ-semiprimary ideal if whenever a,b,c∈R and 0≠abc∈I, then either ab∈δ(I) or bc∈δ(I) or ac∈δ(I).

We start with trivial relations, hence we omit the proof.

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=ℤ210 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 3⋅5⋅14∈I but neither 3⋅5∈δ(I) nor 3⋅14∈δ(I) nor 5⋅14∈δ(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 YZ∈δ1(I) nor XZ∈δ1(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 (0≠abc∈I) abc∈I and ab∉δ(I). Since I⊆δ(I) and δ(I) is 2-absorbing, we have ac∈δ(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,c∈R with abc∈I and ab∉δ(I). Then anbncn∈I for some positive integer n ≥ 1. Since δ(I)⊆δ(I), we conclude anbn∉δ(I). Since I is 2-absorbing δ-semiprimary, we have bncn∈δ(I) or ancn∈δ(I). Since δ(I)⊆δ(I), we conclude that bc∈δ(I) or ac∈δ(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=ℤ12[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(n≥2) be ideals of R such that at most two of them are not 2-absorbing δ1-semiprimary. If I⊆I1∪I2∪⋯∪In, then I⊆Ii for some 1≤i≤n.

Proof. From the covering I⊆I1∪I2∪⋯∪In, we conclude that I⊆I1∪I2∪⋯∪In. By our hypothesis, we may assume that I_k is a 2-absorbing δ1-semiprimary ideal for all k≥3. Hence Ik is 2-absorbing for all k≥3 by Theorem 2.6. Thus I=I⊆Ii=Ii for some 1≤i≤n by [12, Theorem 3.1].

Let I,I1,I2,...,In be ideals of R. Recall that an efficient covering of R is a covering I⊆I1∪I2∪⋯∪In in which no I_j where 1≤j≤n satisfies I⊆Ij (i.e., no I_j 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(n≥2) be ideals of R such that at most two of them not 2-absorbing δ1-semiprimary. Suppose that Ii⊈(Ij:x) for all x∈R\Ij for all i≠j. If I⊆I1∪I2∪⋯∪In, then I⊆Ii for some 1≤i≤n.

Proof. Assume on the contrary that I⊈Ii for all 1≤i≤n. Hence I⊆I1∪I2∪⋯∪In is an efficient covering of ideals of R. So, it is clear that I=(I∩I1)∪(I∩I2)∪⋯∪(I∩In) is an efficient union. From [11, Lemma 2.1], ∩ i≠k(I∩Ii)⊆I∩Ik. Since at most two of the I_i are not 2-absorbing δ1-semiprimary ideals, we may assume that I_k 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 x∉Ii by [4, Theorem 2.5.]. On the other hand, our hypotesis implies that there exist ai∈Ii\(Ik:x) for all i≠ k. Hence there are positive integers m_i such that aimi∈Ii. Put a=∏i≠kai and m=max{m1,m2,...,mn}. It is clear that amx∈∩i≠k(I∩Ii). We show that amx∉I∩Ik. Assume that amx∈I∩Ik. Hence amx∈Ik, and so am∈(Ik:x). Since (Ik:x) is prime, we conclude that ai∈(Ik:x) for some i≠k, a contradiction. Thus ∩i≠k(I∩Ii)\(I∩Ik) is nonempty which contradicts with ∩i≠k(I∩Ii)⊆I∩Ik Thus I⊆Ii for some 1≤i≤n.

Recall from [14] that an ideal expansion δ of I(R) is said to be intersection preserving if it satisfies δ(I1∩I2∩⋯∩In)=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 abc∈I, ab∉δ(I) and bc∉δ(I) for some a,b,c∈R. Since δ(I)=δ(∩ i=1nIi)=∩ i=1nδ(Ii)=K, we have ab∉K and bc∉K. Since abc∈Ii and I_i is 2-absorbing δ-semiprimary, ac∈δ(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 J⊆I and δ(J)=δ(I), then J is a (weakly) 2-absorbing δ-semiprimary ideal of R.

• 2. If J⊆I⊆0, then J is a (weakly) 2-absorbing quasi-primary ideal of R.

• 3. Let I⊆0. If K is an ideal of R, then IK, I∩K and Iⁿ are (weakly) 2-absorbing quasi-primary ideals of R for all positive integers n≥1

Proof. (1) Suppose that (0≠abc∈J)abc∈J for some a,b,c∈R. Since J⊆I, we have (0≠abc∈I) abc∈I. Since I is (weakly) 2-absorbing δ-semiprimary, we have either ab∈δ(I) or bc∈δ(I) or ac∈δ(I). Since δ(I)=δ(J), J is a (weakly) 2-absorbing δ-semiprimary ideal of R.

(2) Since J⊆I⊆0, 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 ab∈δ(abc)⊆δ(I) or bc∈δ(abc)⊆δ(I) or ac∈δ(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 a∈I. Then there exists n which is the least positive integer n with an∈I. If n=1, then a∈I⊆δ(I). For n≥2, an=an−2aa∈I. Since an−1∉I, we have a2∈δ(I). Since δ(I) is semiprime, a∈δ(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 ab∈δ(I) nor bc∈δ(I) nor ac∈δ(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 I⊆0.

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 ab∈δ(I) nor bc∈δ(I) nor ac∈δ(I). for some b,c∈R. First, we show that abI=0. Assume that abi≠0 for some i∈ I. Since 0≠ab(c+i)∈I and I is a weakly 2-absorbing δ-semiprimary, we conclude ab∈I 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 ai1i2≠0 for some i1,i2∈I. Since abI=bcI=acI=0, we have 0≠a(b+i1)(c+i2)=ai1i2∈I. Since I is a weakly 2-absorbing δ-semiprimary, this contradicts our assumption that ab∈δ(I) nor bc∈δ(I) nor ac∈δ(I). Thus aI2=0. One can easily show symmetrically that bI2=cI2=0. Lastly, we show that I3=0. Assume that i1i2i3≠0 for some i1,i2,i3∈I. Since abI=bcI=acI=aI2=bI2=cI2=0, observe that 0≠(a+i1)(b+i2)(c+i2)=i1i2i3∈I. 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=ℤ60 and I=30R. Then I³=0. However, I is not a weakly 2-absorbing δ-semiprimary ideal (for δ=δ0 or δ=δ1) since 2⋅3⋅5∈I but neither 2⋅3∈δ(I) nor 3⋅5∈δ(I) nor 2⋅5∈δ(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 ab∉δ(I), ac∉δ(I) and bc∉δ(I) for some a,b,c∈R. Since δ(I)=δ(0),(a,b,c) is a δ-triple zero of 0. The converse part is obvious.

In particular, suppose that I⊆0 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=ℤ24, 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 2⋅3⋅4=0 but neither 2⋅3∈δ(0) nor 2⋅4∈δ(0) nor 3⋅4∈δ(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 I_i 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 I3⊆0⊆δ(0) by Theorem 2.17. Since δ(0) is a 2-absorbing ideal of R by Theorem , we conclude that I2⊆δ(0) by [4, Theorem 2.13.]. Since 0⊆I2⊆δ(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:R→S is called a δγ-homomorphism if δ(f−1(I))=f−1(γ(I)) for all ideals I of S For example, if γ1 is 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:R→S 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 f−1J 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 f−1J 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 abc∈f−1(J) for some a,b,c∈R. 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 ab∈f−1(γ(J)) or bc∈f−1(γ(J)) or ac∈f−1(γ(J)). Since f−1(γ(J))=δ(f−1(J)),f−1(J) is a 2-absorbing δ-semiprimary ideal of R.

(2) Let 0≠abc∈f−1(J) for some a,b,c∈R. Then f(abc)=f(a)f(b)f(c)∈J. If f(abc)≠0, it can be easily proved similar to (1) that f−1(J) is a weakly 2-absorbing δ-semiprimary ideal of R. Assume that f(abc)=0. Hence abc∈ker(f). Since ker(f) is weakly 2-absorbing δ-semiprimary, we have ab∈δ(ker(f)) or bc∈δ(ker(f)) or ac∈δ(ker(f)). Sinceδ(ker(f))=δ(f−1(0))⊆δ(f−1(J)), we are done.

(3) Let (0≠xyz∈f(I)) xyz∈f(I) for some x,y,z∈S. Then there are some elements a,b,c∈I 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 (0≠abc∈I) abc ∈ I. Since I is (weakly) 2-absorbing δ-semiprimary, we have either ab∈δ(I) or bc∈δ(I) or ac∈δ(I). Thus xy∈f(δ(I)) or yz∈f(δ(I)) or yz∈f(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/I→R/I defined by δq(J/I)=δ(J)/I for all ideals I⊆J, becomes an expansion function of R/I [3]. Consider the natural homomorphism π:R→R/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 I⊆J. Then J is a 2-absorbing δ-semiprimary ideal of R if and only if J/I is a 2-absorbing δ_q-semiprimary ideal of R/I.

• 2. If I is a 2-absorbing δ-semiprimary ideal of R and R′ is a subring with R′⊈I, then I∩R′ 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 δ_q-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 δ_q-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 δ_S is an expansion function of I(RS) such that δS(IS)=(δ(I))S. In the next theorem, we investigate 2-absorbing δ_S-semiprimary ideals of the localization R_S.

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 I∩S=∅, then I_S is a (weakly) 2-absorbing δ_S-semiprimary ideal of R_S.

Proof. Let (0≠xs1ys2zs3∈IS) xs1ys2zs3∈IS for some x,y,z∈R; s1,s2,s3∈S. Then we have (0≠sxyz∈I) sxyz∈ I for some s∈ S. Then sxy∈δ(I) or yz∈δ(I) or sxz∈δ(I). Hence sxyss1s2∈δ(I)S or yzs2s3∈δ(I)S or sxzss1s3∈δ(I)S. Since (δ(I))S=δS(IS), I_S is a (weakly) 2-absorbing δ_S-semiprimary ideal of R_S.

Let R=R1×⋯×Rn(n≥2) 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 abc∈I1 and ab∉δ1(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 bc∈δ1(I1) or ac∈δ1(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 ab∈I1 and a∉δ1(I1) for some a,b∈R1. 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 b∈δ1(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 δ_1,2-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 a1b1∉δ1(I1) and a1c1∉δ1(I1) .Since a1b1c1∈I1, it contradicts with the assumption that I₁ is a δ1-semiprimary ideal.

Case 2: Let a2b2∉δ2(I2) and a2c2∉δ2(I2). Since a2b2c2∈I2, it contradicts with the assumption that I₂ is a δ₂-semiprimary ideal.

Case 3: Let a1b1∉δ1(I1) and a2c2∉δ2(I2). Since a1b1c1∈I1 and I₁ is δ1-semiprimary, we have c1∈δ1(I1). Since a2b2c2∈I2 and I₂ is δ₂-semiprimary, we have b2∈δ2(I2). Thus (b1,b2)(c1,c2)∈δ×(I). textbf{Case 4: }Let a1c1∉δ1(I1) and a2b2∉δ2(I2). Since a1b1c1∈I1 and I₁ is δ1-semiprimary, we have b1∈δ1(I1). Since a2b2c2∈I2 and I₂ is δ₂-semiprimary, we have c2∈δ2(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 n≥2. 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 I_k is a 2-absorbing δ×-semiprimary ideal of R_k and δj(Ij)=Rj for all j∈{1,...,n}\{k} or I_k and I_t are δk,t-semiprimary ideals of R_k and R_t, 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 I3≠0, 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)=R   if 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 I_k is a 2-absorbing δ×-semiprimary ideal of R_k and Ij=Rj for all j∈{1,...,n}\{k} or I_k and I_t are δk,t-semiprimary ideals of R_k and R_t, 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 1∈δ(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):r∈R,m∈M} 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,s∈R; m,m′∈M. 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 IM⊆N ([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 r1r2r3∈I. Since I is 2-absorbing δ-semiprimary, we have r1r2∈δ(I) or r2r3∈δ(I) or r1r3∈δ(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 r1r2r3∈I for some r1,r2,r3∈R. Then (r1,0),(r2,0)(r3,0)=(r1r2r3,0)∈I(+)N. The remain of the proof is clear.