The rings considered in this paper are finite commutative rings with identity, which are not fields. For any ring [Formula: see text] which is not a field and which is not necessarily finite, we denote the set of all zero-divisors of [Formula: see text] by [Formula: see text] and [Formula: see text] by [Formula: see text]. Let [Formula: see text] denote the zero-divisor graph of [Formula: see text] and for a finite ring [Formula: see text], let [Formula: see text] denote the maximum degree of [Formula: see text]. We denote [Formula: see text] by [Formula: see text]. The aim of this paper is to study some properties of [Formula: see text].

In 2005 J. T Wang investigated the zero divisor graphs of degrees 5 and 6. In this paper, we consider the zero divisor graphs of a commutative rings of degrees 7 and 8.

This work aims to introduce and to study a new kind of divisor graph which is  called idempotent divisor graph, and it is  denoted by . Two non-zero distinct vertices v1 and v2 are adjacent if and only if , for some non-unit idempotent element . We establish some fundamental properties of ,  as well as it’s connection with . We also study planarity of this graph.

In this paper several new properties of singular ideals and nonsingular rings are obtained, a connection between a singular ideal and the Jacobson radical is considered , and a sufficient condition for a nonsingular ring to be reduced is given.

In this work, we study generalized simple P-injective rings. New properties of such rings are given, and a characterization of division rings and strongly regular rings in terms of GSP rings is obtained.