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For commutative rings with identity, we introduce and study the concept of semi r-ideals which is a kind of generalization of both r-ideals and semiprime ideals. A proper ideal I of a commutative ring R is called semi r-ideal if whenever a² ∈ I and AnnR(a) = 0, then a ∈ I. Several properties and characterizations of this class of ideals are determined. In particular, we investigate semi r-ideal under various contexts of constructions such as direct products, localizations, homomorphic images, idealizations and amalagamations rings. We extend semi r-ideals of rings to semi r-submodules of modules and clarify some of their properties. Moreover, we define submodules satisfying the D-annihilator condition and justify when they are semi r-submodules. [ABSTRACT FROM AUTHOR]

The concept of 2-nil ideal of a commutative ring R is introduced, as an alternative to the (2, n)-ideals from [6]. We study its relationship with previously introduced classes of ideals, such as 2-absorbing ideals and n-ideals. Various properties and examples are presented. Our main result is a characterization of rings for which every 2-nil ideal is prime. We give a description of 2-nil ideals of general ZPI-rings. [ABSTRACT FROM AUTHOR]

Let R 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 I of R is called a 1 -absorbing primary ideal of R if whenever nonunit elements a , b , c ∈ R and a b c ∈ I , then a b ∈ I or c ∈ I. Some properties of 1-absorbing primary ideals are investigated. For example, we show that if R admits a 1-absorbing primary ideal that is not a primary ideal, then R is a quasilocal ring. We give an example of a 1-absorbing primary ideal of R that is not a primary ideal of R. We show that if R is a Noetherian domain, then R is a Dedekind domain if and only if every nonzero proper 1-absorbing primary ideal of R is of the form P n for some nonzero prime ideal P of R and a positive integer n ≥ 1. We show that a proper ideal I of R is a 1-absorbing primary ideal of R if and only if whenever I 1 I 2 I 3 ⊆ I for some proper ideals I 1 , I 2 , I 3 of R , then I 1 I 2 ⊆ I or I 3 ⊆ I. [ABSTRACT FROM AUTHOR]

A proper element q of a lattice L is said to be a quasi n-absorbing element if whenever anb ⩽ q implies that either an ⩽ q or an-1b ⩽ q. We investigate properties of this new type of elements and obtain some relations among prime, 2-absorbing, n-absorbing elements in multiplicative lattices. [ABSTRACT FROM AUTHOR]