Abstract
In this paper, we introduce and investigate a new graph of a commutative ring R, denoted by G(R), with all nontrivial ideals of R as vertices, and two distinct vertices I and J are adjacent if and only if ann(I∩J)=ann(I)+ann(J). In this article, the basic properties and possible structures of the graph G(R) are studied and investigated as diameter, girth, clique number, cut vertex and domination number. We characterize all rings R for which G(R) is planar, complete and complete r-partite. We show that, if (R,M) is a local Artinian ring, then G(R) is complete if and only if Soc(R) is simple. Also, it is shown that if R is a ring with G(R) is r-regular, then either G(R) is complete or null graph. Moreover, we show that if R is an Artinian ring, then R is a serial ring if and only if G(R/I) is complete for each ideal I of R.
Highlights
Over the last years, there has been an explosion of interest in associating a graph to an algebraic structure
We introduce and investigate a new graph of a commutative ring R, denoted by G(R), with all nontrivial ideals of R as vertices, and two distinct vertices I and J are adjacent if and only if ann(I \ J) = ann(I) + ann(J)
In 1988, Istvan Beck proposed the study of commutative rings by representing them as graphs [6]
Summary
There has been an explosion of interest in associating a graph to an algebraic structure. The following statements are equivalent: (1) G(R) contains an end vertex; (2) Either R = M1 M2, where I(R) = fM1; M2g or (R; M ) is a local ring and each proper non-maximal ideal of R is minimal; (3) G(R) is a star graph; (4) gr(G(R)) = 1; (5) G(R) is a bipartite graph. If I is a maximal ideal of R, jI(R)j = 2, because deg(I) = 1 and I is adjacent to every other vertex of G(R) by Lemma 5.
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