On a graph of ideals of a commutative ring

Authors : Mehdi KHORAMDEL, Shahabaddin EBRAHİMİ ATANİ, Saboura DOLATİ PİSHHESARİ

Pages : 2283-2297

Doi:10.31801/cfsuasmas.534944

View : 23 | Download : 11

Publication Date : 2019-08-01

Article Type : Research Paper

Abstract :In this paper, we introduce and investigate a new graph of a commutative ring R, denoted by Ginsert ignore into journalissuearticles values(R);, with all nontrivial ideals of R as vertices, and two distinct vertices I and J are adjacent if and only if anninsert ignore into journalissuearticles values(I∩J);=anninsert ignore into journalissuearticles values(I);+anninsert ignore into journalissuearticles values(J);. In this article, the basic properties and possible structures of the graph Ginsert ignore into journalissuearticles values(R); are studied and investigated as diameter, girth, clique number, cut vertex and domination number. We characterize all rings R for which Ginsert ignore into journalissuearticles values(R); is planar, complete and complete r-partite. We show that, if insert ignore into journalissuearticles values(R,M); is a local Artinian ring, then Ginsert ignore into journalissuearticles values(R); is complete if and only if Socinsert ignore into journalissuearticles values(R); is simple. Also, it is shown that if R is a ring with Ginsert ignore into journalissuearticles values(R); is r-regular, then either Ginsert ignore into journalissuearticles values(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 Ginsert ignore into journalissuearticles values(R/I); is complete for each ideal I of R.
Keywords : Ideals, clique number, complete r partite, planar property