Arithmetical Semigroup Rings
Thomas Shores1980, Canadian Journal of Mathematics
Sign up for access to the world's latest research
Abstract
Throughout this paper the ring R and the semigroup S are commutative with identity; moreover, it is assumed that S is cancellative, i.e., that S can be embedded in a group. The aim of this note is to determine necessary and sufficient conditions on R and 5 that the semigroup ring R[S] should be one of the following types of rings: principal ideal ring (PIR), ZPI-ring, Bezout, semihereditary or arithmetical. These results shed some light on the structure of semigroup rings and provide a source of examples of the rings listed above. They also play a key role in the determination of all commutative reduced arithmetical semigroup rings (without the cancellative hypothesis on S) which will appear in a forthcoming paper by Leo Chouinard and the authors [4]. Our results are motivated in large part by the paper [11] of R. Gilmer and T. Parker. In particular, Theorem 1.1 of [11] asserts that if R and S are as above and, moreover, if 5 is torsion-free, then the following are equivalent conditions: (1) R[S] is a Bezout ring; (2) R[S] is a Priifer ring; (3) R is a (von Neumann) regular ring and 5 is isomorphic to either a subgroup of the additive rationals or the positive cone of such a subgroup. One could very naturally include a fourth condition, namely: (4) R[S] is arithmetical. L. Fuchs [7] defines an arithmetical ring as a commutative ring with identity for which the ideals form a distributive lattice. Since a Priifer ring is one for which (A + B) C\ C = {A C\ C) + (B Pi C) whenever at least one of the ideals A, B or C contains a regular element (see [18]), arithmetical rings are certainly Priifer. On the other hand, it is well known that every Bezout ring is arithmetical, so that (4) is indeed equivalent to (l)-(3) in Theorem 1.1. In Theorem 3.6 of this paper we drop the requirement that S be torsion-free and determine necessary and sufficient conditions for the semigroup ring of a cancellative semigroup to be arithmetical. Examples are included to show that for these more general semigroup rings, the equivalences of the torsion-free case are no longer true. Theorems 4.1 and 4.2 provide characterizations of semigroup rings that are ZPI-rings and PIR's. Again, the corresponding results in [18] for torsion-free semigroups fail to hold in the more general case. We would like to thank Leo Chouinard for showing us how to remove
