Prime Decomposition of Zero Divisor Graph in a Commutative Ring

Abstract

Let R be a commutative ring and let $\mathrm{\Gamma }\left(Z_n\right)$ be the zero divisor graph of a commutative ring $R$ , whose vertices are nonzero zero divisors of $Z_n$ , and such that the two vertices $u,v$ are adjacent if $n$ divides $uv$ . In this paper, we introduce the concept of prime decomposition of zero divisor graph in a commutative ring and also discuss some special cases of $\mathrm{\Gamma }\left(Z_{3p}\right)$ , $\mathrm{\Gamma }\left(Z_{5p}\right)$ , $\Gamma \left(Z_{7p}\right)$ , and $\mathrm{\Gamma }\left(Z_{pq}\right)$ .