Group rings and semigroup rings over strong Mori domains, II

Abstract

Let R be an integral domain and let S be a torsion-free cancellative additive monoid with quotient group G. We show that the semigroup ring R[X; S] is a strong Mori domain if and only if R is a strong Mori domain, S is a strong Mori semigroup, and each nonzero element of G is of type (0, 0, 0, . . .).  2004 Elsevier Inc. All rights reserved.For an associative ring R and a semigroup S (written additively), N. Jacobson [18, p. 95] defines the semigroup ring of S over R to be the set of functions f from S into R that are finitely nonzero, where addition and multiplication are defined by the rules(s) = t +u=s f (t)g(u). Following D.G. Northcott [20, p. 128], we denote the semigroup ring of S over R by the symbol R[X; S]; we write the elements of R[X; S] in the form f i X s i , f i ∈ R, s i ∈ S. A general reference on semigroups and semigroup rings is [12]. As we are dealing only with semigroup rings R[X; S] that are integral domains, we impose the corresponding restrictions on R and S (cf. [12, Theorem 8.1]). Namely, we assume that R is an integral domain with quotient field K, S a torsion-free cancellative additive semigroup with 0 (i.e., monoid) and G the quotient group of S.

Arithmetic of Mori domains and monoids

“…Mori domains were introduced in the 1970s [38,39] and attracted a lot of attention since that time (see the works of V. Barucci, S. Gabelli, E. Houston, T.G. Lucas, M. Roitman and others [4][5][6]37,42]). …”