On weakly 2-absorbing δ-primary ideals of commutative rings

Abstract

Abstract Let R be a commutative ring with 1 ≠ 0 {1\neq 0} . We recall that a proper ideal I of R is called a weakly 2-absorbing primary ideal of R if whenever a , b , c ∈ R {a,b,c\in R} and 0 ≠ a ⁢ b ⁢ c ∈ I {0\not=abc\in I} , then a ⁢ b ∈ I {ab\in I} or a ⁢ c ∈ I {ac\in\sqrt{I}} or b ⁢ c ∈ I {bc\in\sqrt{I}} . In this paper, we introduce a new class of ideals that is closely related to the class of weakly 2-absorbing primary ideals. Let I ⁢ ( R ) {I(R)} be the set of all ideals of R and let δ : I ⁢ ( R ) → I ⁢ ( R ) {\delta:I(R)\rightarrow I(R)} be a function. Then δ is called an expansion function of ideals of R if whenever L , I , J {L,I,J} are ideals of R with J ⊆ I {J\subseteq I} , then L ⊆ δ ⁢ ( L ) {L\subseteq\delta(L)} and δ ⁢ ( J ) ⊆ δ ⁢ ( I ) {\delta(J)\subseteq\delta(I)} . Let δ be an expansion function of ideals of R. Then a proper ideal I of R (i.e., I ≠ R {I\not=R} ) is called a weakly 2-absorbing δ-primary ideal if 0 ≠ a ⁢ b ⁢ c ∈ I {0\not=abc\in I} implies a ⁢ b ∈ I {ab\in I} or a ⁢ c ∈ δ ⁢ ( I ) {ac\in\delta(I)} or b ⁢ c ∈ δ ⁢ ( I ) {bc\in\delta(I)} . For example, let δ : I ⁢ ( R ) → I ⁢ ( R ) {\delta:I(R)\rightarrow I(R)} such that δ ⁢ ( I ) = I {\delta(I)=\sqrt{I}} . Then δ is an expansion function of ideals of R, and hence a proper ideal I of R is a weakly 2-absorbing primary ideal of R if and only if I is a weakly 2-absorbing δ-primary ideal of R. A number of results concerning weakly 2-absorbing δ-primary ideals and examples of weakly 2-absorbing δ-primary ideals are given.