Abstract Let R be a finite commutative ring. We define co-unit graph, associated to ring , denoted by G nu ( ) with vertex set V )) = U ), where is the of units and two distinct vertices x y being adjacent if only + ∉ / ). In this paper, we investigate some basic properties integers modulo n, for different values n. find domination number, clique number girth

Let ?: R0?R be a ring homomorphism and suppose that a and a0, respectively, are ideals of R and R0 such that is an Artinian ring. Let M and N be two finitely generated R-modules and suppose that (R0,m0) is a local ring. In this note we prove that the R-modules and are Artinian for all integers i and j, whenever and . Also we will show that if a is principal, then the R-modules and ...

A unit x in a commutative ring R with identity is called exceptional if 1−x is also a unit in R. For any integer n ≥ 2, define φe(n) to be the number of exceptional units in the ring of integers modulo n. Following work of Shapiro, Mills, Catlin and Noe on iterations of Euler’s φ-function, we develop analogous results on iterations of the function φe, when restricted to a particular subset of t...

In this paper we study the structure and the commutativity of a ring R, in which for each x,y ? R, there exist two integers depending on x,y such that [x,y]k equals x n or y n.

The binary operation multint on Z is defined as follows: (Def. 1) For all elements a, b of Z holds (multint)(a, b) = ·R(a, b). The unary operation compint on Z is defined as follows: (Def. 2) For every element a of Z holds (compint)(a) = −R(a). The double loop structure INT.Ring is defined by: (Def. 3) INT.Ring = 〈Z, +Z,multint, 1(∈ Z), 0(∈ Z)〉. Let us mention that INT.Ring is strict and non em...

in this paper we study the structure and the commutativity of a ring r, in which for each x,y ? r, there exist two integers depending on x,y such that [x,y]k equals x n or y n.

Given a commutative ring R with identity 1?0, let the set Z(R) denote of zero-divisors and Z*(R)=Z(R)?{0} be non-zero R. The zero-divisor graph R, denoted by ?(R), is simple whose vertex Z*(R) each pair vertices in are adjacent when their product 0. In this article, we find structure Laplacian spectrum graphs ?(Zn) for n=pN1qN2, where p<q primes N1,N2 positive integers.