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Cycles and symmetries of zero-divisors. (English) Zbl 1087.13500
From the introduction: Let R be a commutative ring with 10 and let Z(R) denote the set of nonzero zero-divisors of R. By the zero-divisor-graph of R we mean the graph with vertices Z(R) such that there is an (undirected) edge between vertices x,y if and only if xy and xy=0. A subset of Z(R) determines a subgraph of the zero-divisor-graph. In this article we investigate the nature of cycles (circuits) in a zero-divisor graph. There is a canonical decomposition of the graph where each component is easy to describe.

MSC:
13A05Divisibility
05C38Paths; cycles
05C90Applications of graph theory