THE TOTAL GRAPH OF A COMMUTATIVE RING WITH RESPECT TO PROPER IDEALS

Title & Authors
THE TOTAL GRAPH OF A COMMUTATIVE RING WITH RESPECT TO PROPER IDEALS

Abstract
Let R be a commutative ring and I its proper ideal, let S(I) be the set of all elements of R that are not prime to I. Here we introduce and study the total graph of a commutative ring R with respect to proper ideal I, denoted by T($\small{{\Gamma}_I(R)}$). It is the (undirected) graph with all elements of R as vertices, and for distinct x, y $\small{{\in}}$ R, the vertices x and y are adjacent if and only if x + y $\small{{\in}}$ S(I). The total graph of a commutative ring, that denoted by T($\small{{\Gamma}(R)}$), is the graph where the vertices are all elements of R and where there is an undirected edge between two distinct vertices x and y if and only if x + y $\small{{\in}}$ Z(R) which is due to Anderson and Badawi [2]. In the case I = {0}, $\small{T({\Gamma}_I(R))=T({\Gamma}(R))}$; this is an important result on the definition.
Keywords
commutative rings;zero divisor;total graph;
Language
English
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