GRAPHS ASSOCIATED TO CO-MAXIMAL IDEALS OF COMMUTATIVE RINGS

By Hsin-Ju Wang. 2008

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Abstract Let R be a commutative ring R with 1. In [P.K. Sharma, S.M. Bhatwadekar, A note on graphical representation of rings, J. Algebra 176 (1995) 124–127], Sharma and Bhatwadekar defined a graph on R, Γ ( R ) , with vertices as elements of R, where two distinct vertices a and b are adjacent if and only if a R + b R = R . Let Γ 2 ( R ) be the subgraph of Γ ( R ) induced by non-unit elements. In this paper, we derive several properties of Γ 2 ( R ) . We are able to characterize those rings R for which Γ 2 ( R ) − J ( R ) is a forest and those rings R for which Γ 2 ( R ) − J ( R ) is Eulerian, where J ( R ) is the Jacobson radical of R. We are also able to show that ω ( Γ 2 ( R ) ) = | Max ( R ) | , where ω ( G ) is the clique number of a graph G and Max ( R ) is the set of maximal ideals of R. Moreover, we find all finite rings R such that the genus of Γ 2 ( R ) (resp. Γ ( R ) ) is at most one.

Subfields: Algebra and Number Theory

View Paper: https://www.semanticscholar.org/paper/00aa9ca7d5ee4e915b6b89618f6ad9fa7c27fa3f

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In Algebra and Number Theory

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