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Cut-sets in zero-divisor graphs of finite commutative rings. (English) Zbl 1228.13011

Let \(R\) be a commutative ring with nonzero identity, and let \(Z(R)\) be its set of zero-divisors. The zero-divisor graph, \(\Gamma(R)\), is the graph with vertices the set of nonzero zero-divisors of \(R\), and for distinct \(x,y \in {Z(R)}\), the vertices \(x\) and \(y\) are adjacent if and only if \(xy=0\). The zero-divisor graph of a commutative ring has been studied extensively by several authors. For an arbitrary connected graph \(G\) with the set of vertices \(V(G)\), a set \(A\subset V(G)\) is said to be a cut-set if there exist distinct \(c, d\in V(G)\setminus A\) such that every path in \(G\) from \(c\) to \(d\) involves at least one element of \(A\), and no proper subset of \(A\) satisfies the same condition.
In the paper under review the authors examine minimal sets of vertices which, when removed from a zero-divisor graph, separate the graph into disconnected subgraphs. In addition, the authors classify these sets for all finite nonlocal commutative rings with identity.

MSC:

13A99 General commutative ring theory

Software:

Mathematica
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References:

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