On generalized power series rings with some restrictions on zero-divisors

Abstract

Let [Formula: see text] be a ring and [Formula: see text] a strictly ordered monoid. The construction of generalized power series ring [Formula: see text] 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 [Formula: see text] and a m.a.n.u.p. monoid [Formula: see text], it is shown that (i) [Formula: see text] is power-serieswise [Formula: see text]-McCoy, (ii) [Formula: see text] have Property (A), (iii) [Formula: see text] is right zip if and only if [Formula: see text] is right zip, (iv) [Formula: see text] is strongly AB if and only if [Formula: see text] is strongly AB. Also we study the interplay between ring-theoretical properties of a generalized power series ring [Formula: see text] and the graph-theoretical properties of its undirected zero divisor graph of [Formula: see text]. A complete characterization for the possible diameters [Formula: see text] is given exclusively in terms of the ideals of [Formula: see text]. Also, we present some examples to show that the assumption “R is right Noetherian” in our main results is not superfluous.