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Zero-divisors and zero-divisor graphs of power series rings. (English) Zbl 1346.13044

Authors’ abstract: Let \(R\) be a commutative ring with identity, \(Z(R)\) its set of zero-divisors and \(N(R)\) its nilradical. The zero-divisor graph of \(R\) denoted by \(\Gamma(R)\), is the graph with vertices \(Z(R)\backslash(0)\), and with distinct adjacent vertices \(x\) and \(y\) if and only if \(xy=0\). In this paper we give some results about zero-divisors in the power series ring \(R[[X]]\), and we study the diameter of \(\Gamma(R[[X]])\) in the case when \(N(R)=Z(R)\). We also give some results when \(N(R)\subsetneq Z(R)\). Among these case, we prove that \(\mathrm{diam}(\Gamma(R))=\mathrm{diam}(\Gamma(R[X]))=2\) and \(\mathrm{diam}(\Gamma(R[[X]]))=3\) if \(R\) is one of the following rings: divided ring, PVR ring, chained ring, or \(R\) is a ring such that \(Z(R)=aR+I\) with \(a\in Z(R)\backslash N(R)\) and \(I\subsetneq (0:a)\).

MSC:

13F25 Formal power series rings
13A99 General commutative ring theory
13J05 Power series rings
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References:

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