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.

‎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 $R$‎.

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.

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.

Let R be a Commutative ring with identity. We use j (R) to denote the comaximal ideal graph. The vertices of j (R) are proper ideals of R which are not contained in the Jacobson radical of R, and two vertices I and J are adjacent if and only if I+J=R. In this paper we show some properties of this graph together with planarity of line graph associated to j (R).

Let $\mathcal{B}$ be a Commutative ring with unity $1\neq 0$, $1\leq m <\infty$ be an integer and $\mathcal{R}=\mathcal{B}\times \mathcal{B} \times\cdots\times \mathcal{B}$ ($m$ times). The total essential dot product graph $ETD(\mathcal{R})$ and the essential zero-divisor dot product graph $EZD(\mathcal{R})$ are undirected graphs with the vertex sets $\mathcal{R}^{*} = \mathcal{R}\setminus \{(0,0,...0)\}$ and $Z(\mathcal{R})^*=Z(\mathcal{R})\setminus \{(0,0,...,0)\}$ respectively. Two distinct vertices $w=(w_1,w_2,...,w_m)$ and $z=(z_1,z_2,...,z_m)$ are adjacent if and only if $ann_\mathcal{B}(w\cdot z)$ is an essential ideal of $\mathcal{B}$ (where $w\cdot z=w_1z_1+w_2z_2+\cdots +w_mz_m\in \mathcal{B}$). In this paper, we prove some results on connectedness, diameter and girth of $ETD(\mathcal{R})$ and $EZD(\mathcal{R})$. We classify the ring $\mathcal{R}$ such that $EZD(\mathcal{R})$ and $ETD(\mathcal{R})$ are planar, outerplanar, and of genus one.