Search

Let Rbe a commutative ring with identity, and let Sbe a multiplicative subset of R. Then R is called a nonnil-S-Noetherian ring if every nonnil ideal of Ris S-finite. Also, Ris called a u-S-Noetherian ring if there exists an element s∈ Ssuch that for each ideal Iof R, sI⊆ Kfor some finitely generated sub-ideal Kof I. In this paper, we examine some new characterization of nonnil-S-Noetherian rings. Then, as a generalization of nonnil-S-Noetherian rings and u-S-Noetherian rings, we introduce and investigate the nonnilu-S-Noetherian rings class.

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 Iof a commutative ring Ris called semi r-ideal if whenever a2∈ Iand 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.

Let Rbe a commutative ring with non-zero identity and Mbe a unitary R-module. The goal of this paper is to extend the concept of 1-absorbing primary ideals to 1-absorbing primary submodules. A proper submodule Nof Mis said to be a 1-absorbing primary submodule if whenever non-unit elements a, b∈ Rand m∈ Mwith abm∈ N, then either ab∈ (N:RM) or m∈ M− rad(N). Various properties and chacterizations of this class of submodules are considered. Moreover, 1-absorbing primary avoidance theorem is proved.

Let G be a group with identity e and let R be a G-graded ring. In this paper, we introduce and study the concept of gr-n-ideals of R. We obtain many results concerning gr-n-ideals. Some characterizations of gr-n-ideals and their homogeneous components are given.