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

- Journal title : Journal of the Korean Mathematical Society
- Volume 49, Issue 1, 2012, pp.85-98
- Publisher : The Korean Mathematical Society
- DOI : 10.4134/JKMS.2012.49.1.085

Title & Authors

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

Abbasi, Ahmad; Habibi, Shokoofe;

Abbasi, Ahmad; Habibi, Shokoofe;

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(). It is the (undirected) graph with all elements of R as vertices, and for distinct x, y R, the vertices x and y are adjacent if and only if x + y S(I). The total graph of a commutative ring, that denoted by T(), 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 Z(R) which is due to Anderson and Badawi [2]. In the case I = {0}, ; this is an important result on the definition.

Keywords

commutative rings;zero divisor;total graph;

Language

English

Cited by

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TOTAL GRAPH OF A COMMUTATIVE SEMIRING WITH RESPECT TO IDENTITY-SUMMAND ELEMENTS,;;;

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THE TOTAL TORSION ELEMENT GRAPH WITHOUT THE ZERO ELEMENT OF MODULES OVER COMMUTATIVE RINGS,;

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