Let [Formula: see text] be a commutative ring with nonzero identity. In this paper, we introduce the concept of 1-absorbing primary ideals in commutative rings. A proper ideal [Formula: see text] of [Formula: see text] is called a [Formula: see text]-absorbing primary ideal of [Formula: see text] if whenever nonunit elements [Formula: see text] and [Formula: see text], then [Formula: see text] or [Formula: see text] Some properties of 1-absorbing primary ideals are investigated. For example, we show that if [Formula: see text] admits a 1-absorbing primary ideal that is not a primary ideal, then [Formula: see text] is a quasilocal ring. We give an example of a 1-absorbing primary ideal of [Formula: see text] that is not a primary ideal of [Formula: see text]. We show that if [Formula: see text] is a Noetherian domain, then [Formula: see text] is a Dedekind domain if and only if every nonzero proper 1-absorbing primary ideal of [Formula: see text] is of the form [Formula: see text] for some nonzero prime ideal [Formula: see text] of [Formula: see text] and a positive integer [Formula: see text]. We show that a proper ideal [Formula: see text] of [Formula: see text] is a 1-absorbing primary ideal of [Formula: see text] if and only if whenever [Formula: see text] for some proper ideals [Formula: see text] of [Formula: see text], then [Formula: see text] or [Formula: see text]

Let R be a commutative ring with 1 ̸ = 0 and S(R) be the set of all ideals of R . In this paper, we extend the concept of 2-absorbing primary ideals to the context of ϕ -2-absorbing primary ideals. Let ϕ : S(R) → S(R) ∪ ∅ be a function. A proper ideal I of R is said to be a ϕ -2-absorbing primary ideal of R if whenever a, b, c ∈ R with abc ∈ I − ϕ(I) implies ab ∈ I or ac ∈ √ I or bc ∈ √ I . A number of results concerning ϕ -2-absorbing primary ideals are given.

In this paper, we introduce and study the concept of graded 2-absorbing primary submodules of graded modules over graded commutative rings generalizing graded 2-absorbing submodules. Let R be a graded ring and M be a graded R-module. A proper graded submodule N of M is called a graded 2-absorbing primary submodule of M if whenever a, b ∈ h(R) and m ∈ h(M ) and abm ∈ N , then am ∈ M -Gr(N ) or bm ∈ M -Gr(N ) or ab ∈ (N :R M ).

Let R be a commutative ring with nonzero identity, let I(R) be the set of all ideals of R and δ : I(R) → I(R) an expansion of ideals of R defined by I → δ(I). We introduce the concept of (δ, 2)-primary ideals in commutative rings. A proper ideal I of R is called a (δ, 2)-primary ideal if whenever a, b ∈ R and ab ∈ I, then a 2 ∈ I or b 2 ∈ δ(I). Our purpose is to extend the concept of 2-ideals to (δ, 2)-primary ideals of commutative rings. Then we investigate the basic properties of (δ, 2)-primary ideals and also discuss the relations among (δ, 2)-primary, δ-primary and 2-prime ideals.

Let [Formula: see text] be a commutative ring with identity. The prime ideal sum graph of [Formula: see text], denoted by [Formula: see text], is a graph whose vertices are nonzero proper ideals of [Formula: see text] and two distinct vertices [Formula: see text] and [Formula: see text] are adjacent if and only if [Formula: see text] is a prime ideal of [Formula: see text]. In this paper, we study some connections between the graph-theoretic properties of this graph and some algebraic properties of rings. The clique number, the chromatic number and the domination number of the prime ideal sum graph for some classes of rings are studied. It is observed that under which condition [Formula: see text] is complete. Moreover, the diameter and the girth of [Formula: see text] are studied.