COMMUTATIVE LOCAL RINGS OF BOUNDED MODULE TYPE

Abstract

Let R be a local ring of bounded module type. It is shown that R is an almost maximal valuation ring if there exists a non-maximal prime ideal J such that R/J is an almost maximal valuation domain. We deduce from this that R is almost maximal if one of the following conditions is satisfied: R is a -algebra with Krull dimension ≥1 or the maximal ideal of R is the union of all non-maximal prime ideals.

Keywords:

• Valuation ring
• Bounded module type
• Completion

Acknowledgments