Semilocal domains whose finitely generated modules are direct sums of cyclics

Abstract

A necessary and sufficient condition is given for a semilocal domain to have the property that every finitely generated module is a direct sum of cyclic modules. Recent work by Tom Shores and Roger Wiegand [SW] and [SWB] has shed new light on the old question: What is the class of commutative rings for which every finitely generated module is a direct sum of cyclic submodules? Following their terminology, such a ring is called an FGC-ring. In this paper, we are able to settle one of their conjectures [SWB, p. 1279], thus giving a characterization of reduced FGC-rings with Noetherian maximal ideal spectrum. We show that, for R a reduced ring with Noetherian maximal ideal spectrum, R is an FGC-ring and only R is a finite direct sum of h-local Bezout domains and each localization of R is an almost maximal valuation ring. The if part of the statement above was shown in [SW] to be an easy consequence of results due to Eben Matlis [M]. Other known results on FGC-rings are reviewed in [SW]; the reader is also referred to [SW] for definitions of the concepts above. The examples constructed in [W] show that our main result answers the FGC question for a large class of rings where it was unknown. Of course it would be nice to give a complete characterization of reduced FGC-rings. This would require settling another question in [SW]: Does every FGC-ring have Noetherian maximal ideal spectrum? They show the answer is yes the ring has fewer than 2' prime ideals [SW, Corollary 6.8]. 1. The main theorem. The result stated in the introduction will easily follow from the Theorem below: Theorem. Let R be an FGC-domain. Then every nonzero prime ideal P is contained in a unique maximal ideal. Proof. Suppose R contains two distinct maximal ideals M and N with Received by the editors May 16, 1974. AMS (MOS) subject classifications (1970). Primary 13C05, 13F05; Secondary 13A15. Copyright ? 1975, Anierican Mathematical Society