Let R be a commutative ring with identity. We associate a graph Γ(R). In this paper, we find Hosoya polynomial and Wiener index of Γ(Zn), with n= p m or n= p m q, where p and q are distinct prime numbers and m is an integer with m≥2.

Let I be a right ideal of R, then R / I is a right N-flat if and only if for each a  I, there exists b  I and a positive integer n such that a n ≠ 0 and a n = ba n . In this paper, we first give and develop various properties of right N-flat rings, by which, many of the known results are extended. Also, we study the relations between such rings and regular, biregular ring.

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].

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 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.