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Directed zero-divisor graph and skew power series rings. (English) Zbl 1463.16072

Summary: Let \(R\) be an associative ring with identity and \(Z^* (R)\) be its set of non-zero zero-divisors. Zero-divisor graphs of rings are well represented in the literature of commutative and non-commutative rings. The directed zero-divisor graph of \(R\), denoted by \(\Gamma (R)\), is the directed graph whose vertices are the set of non-zero zero-divisors of \(R\) and for distinct non-zero zero-divisors \(x\), \(y\), \(x\rightarrow y\) is an directed edge if and only if \(xy=0\). In this paper, we connect some graph-theoretic concepts with algebraic notions, and investigate the interplay between the ring-theoretical properties of a skew power series ring \(R[[x;\alpha]]\) and the graph-theoretical properties of its directed zero-divisor graph \(\Gamma(R[[x;\alpha]])\). In doing so, we give a characterization of the possible diameters of \(\Gamma (R[[x;\alpha]])\) in terms of the diameter of \(\Gamma (R)\), when the base ring \(R\) is reversible and right Noetherian with an \(\alpha\)-condition, namely \(\alpha\)-compatible property. We also provide many examples for showing the necessity of our assumptions.

MSC:

16U99 Conditions on elements
16S36 Ordinary and skew polynomial rings and semigroup rings
05C25 Graphs and abstract algebra (groups, rings, fields, etc.)
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