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

Ayman Badawi 1  and Brahim Fahid 2
• 1 Department of Mathematics & Statistics, American University of Sharjah, P.O. Box 26666, Sharjah, United Arab Emirates
• 2 Department of Mathematics, Mohammed V University, Faculty of Sciences, B.P. 1014, Rabat, Morocco
• Corresponding author
• Department of Mathematics & Statistics, American University of Sharjah, P.O. Box 26666, Sharjah, United Arab Emirates
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and Brahim Fahid

## Abstract

Let R be a commutative ring with $1≠0$. We recall that a proper ideal I of R is called a weakly 2-absorbing primary ideal of R if whenever $a,b,c∈R$ and $0≠a⁢b⁢c∈I$, then $a⁢b∈I$ or $a⁢c∈I$ or $b⁢c∈I$. In this paper, we introduce a new class of ideals that is closely related to the class of weakly 2-absorbing primary ideals. Let $I⁢(R)$ be the set of all ideals of R and let $δ:I⁢(R)→I⁢(R)$ be a function. Then δ is called an expansion function of ideals of R if whenever $L,I,J$ are ideals of R with $J⊆I$, then $L⊆δ⁢(L)$ and $δ⁢(J)⊆δ⁢(I)$. Let δ be an expansion function of ideals of R. Then a proper ideal I of R (i.e., $I≠R$) is called a weakly 2-absorbing δ-primary ideal if $0≠a⁢b⁢c∈I$ implies $a⁢b∈I$ or $a⁢c∈δ⁢(I)$ or $b⁢c∈δ⁢(I)$. For example, let $δ:I⁢(R)→I⁢(R)$ such that $δ⁢(I)=I$. 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.

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