In this paper, we introduce a new class of modules that is closely related to the class of Noetherian modules. Let R be a commutative ring with identity, and M be an R-module such that Nil(M) is a divided prime submodule of M. M is called a nonnil-Noetherian R-module if every nonnil submodule of M is  nitely-generated. We prove that many properties of the Noether-ian modules are also true for the nonnil-Noetherian modules.

Prime graph of a ring R is a graph whose vertex set is the whole set R any any two elements x and y of R are adjacent in the graph if and only if xRy = 0 or yRx = 0. Prime graph of a ring is denoted by PG(R). Directed prime graphs for non-commutative rings and connectivity in the graph are studied in the present paper. The diameter and girth of this graph are also studied in the paper.

The annihilator graph AG (R) of a commutative ring R is a simple undirected graph with the vertex setZ (R)* and two distinct vertices are adjacent if and only if ann (x) È ann (y)¹ann (xy). In this paper we give the sufficient condition for a graph AG (R) to be complete. We characterize rings for which AG (R) is a regular graph, we show that g (AG (R)) Î {1, 2} g and we also characterize the rings for which AG (R) has a cut vertex. Finally we find the clique number of a finite reduced ring and characterize the rings for which AG (R) is a planar graph.

‎Let $R$ be a commutative ring with nonzero identity‎. ‎Throughout this paper we explore some properties of two certain subgraphs of the maximal graph of $R$‎.

In this paper, our research sheds new light on generalized ideals, significantly advancing the state of knowledge in ring theory. We introduce an almost δ-primary ideal which unifies an almost prime ideal and an almost primary ideal. We also define and study the concept of a φ-δ-primary ideal in a commutative ring. Some characterizations of almost δ-primary ideal and n-almost δ-primary ideals are proved.

This paper deals with the number theoretic properties of non-unit elements of the ring 𝑍 𝑛 . Let 𝐷 be the set of all non-trivial divisors of a positive integer 𝑛 . Let 𝐷 1 and 𝐷 2 be the subsets of 𝐷 having the least common multiple which are incongruent to zero modulo 𝑛 with every other element of 𝐷 and congruent to zero modulo 𝑛 with at least one another element of 𝐷 , respectively. Then 𝐷 can be written as the disjoint union of 𝐷 1 and 𝐷 2 in 𝑍 𝑛 . We explore the results on these sets based on all the characterizations of 𝑛 . We obtain a formula for enumerating the cardinality of the set of all non-unit elements in 𝑍 𝑛 whose principal ideals are equal. Further, we present an algorithm for enumerating these sets of all non-unit elements.

In this paper we introduce and study a graph on the set of ideals of a commutative ring RR. The vertices of this graph are non-trivial ideals of RR and two distinct ideals II and JJ are adjacent if and only IJ=I∩ JIJ=I∩ J. We obtain some properties of this graph and study its relation to the structure of RR.

Let R be a commutative ring with non-zero identity. The annihilatorinclusion ideal graph of R, denoted by  R, is a graph whose vertex set is the of all non-zero proper ideals of R and two distinct vertices I and J are adjacent if and only if either Ann(I)  J or Ann(J)  I. The purpose of this paper is to provide some basic properties of the graph  R. In particular, shows that  R is a connected graph with diameter at most three, and has girth 3 or 1. Furthermore, is determined all isomorphic classes of non-local Artinian rings whose annihilator-inclusion ideal graphs have genus zero or one.