Volume 04,Issue 04

Zero Divisor Graphs of Idealizations for Direct Products of Commutative Rings

Authors

Mehrdad Azadi, Azemat Ramzanpoor Mardekheh

Abstract
Let R_1and R_2be commutative ring, M be a prime module over R_1and R_2. Let Z(R) be the set of zero divisors of a commutative ring R. The zero divisor graph of a commutative ring R, is the simple graph with vertices Z(R)-{0}, denoted ?(R). Two distinct vertices x and y of Z(R)-{0} are adjacent if and only if xy=0. In this paper we study the diameter of ?((R_1*R_2) (+) M) with respect to the diameters of the zero divisor graphs of R_1(+) M and R_2(+) M.

Keyword: Idealization, Prime, Module, Zero divisor, Graph, Direct Product.

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