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On generalized power series rings with some restrictions on zero-divisors. (English) Zbl 1415.16033

Summary: Let \(R\) be a ring and \((S,\leq)\) a strictly ordered monoid. The construction of generalized power series ring \(R[[S]]\) generalizes some ring constructions such as polynomial rings, group rings, power series rings and Mal’cev-Neumann construction. In this paper, for a reversible right Noetherian ring \(R\) and a m.a.n.u.p. monoid \((S,\leq)\), it is shown that (i) \(R\) is power-serieswise \(S\)-McCoy, (ii) \(R[[S]]\) have Property (A), (iii) \(R\) is right zip if and only if \(R[[S]]\) is right zip, (iv) \(R\) is strongly AB if and only if \(R[[S]]\) is strongly AB. Also we study the interplay between ring-theoretical properties of a generalized power series ring \(R[[S]]\) and the graph-theoretical properties of its undirected zero divisor graph of \(\Gamma(R[[S]])\). A complete characterization for the possible diameters \(\Gamma(R[[S]])\) is given exclusively in terms of the ideals of \(R\). Also, we present some examples to show that the assumption “\(R\) is right Noetherian” in our main results is not superfluous.

MSC:

16U99 Conditions on elements
13A99 General commutative ring theory
16S99 Associative rings and algebras arising under various constructions
05C12 Distance in graphs
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