Some properties of comaximal right ideal graph of a ring

Shouqiang Shen, Weijun Liu and Lihua Feng

Applied Mathematics and Computation, 2018, vol. 333, issue C, 225-230

Abstract: For a ring R (not necessarily commutative) with identity, the comaximal right ideal graph of R, denoted by G(R), is a graph whose vertices are the nonzero proper right ideals of R, and two distinct vertices I and J are adjacent if and only if I+J=R. In this paper we consider a subgraph G*(R) of G(R) induced by V(G(R))∖J(R), where J(R) is the set of all proper right ideals contained in the Jacobson radical of R. We prove that if R contains a nontrivial central idempotent, then G*(R) is a star graph if and only if R is isomorphic to the direct product of two local rings, and one of these two rings has unique maximal right ideal {0}. In addition, we also show that R has at least two maximal right ideals if and only if G*(R) is connected and its diameter is at most 3, then completely characterize the diameter of this graph.

Keywords: Comaximal right ideal graph; Central idempotent; Local ring (search for similar items in EconPapers)
Date: 2018
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Persistent link: https://EconPapers.repec.org/RePEc:eee:apmaco:v:333:y:2018:i:c:p:225-230

DOI: 10.1016/j.amc.2018.03.075

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