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2001•Communications in AlgebraRequires access

COMMUTATIVE LOCAL RINGS OF BOUNDED MODULE TYPE

François Couchot

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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.

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What this paper is about

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.

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Available 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.

Key concepts: Mathematics, Maximal ideal, Krull dimension, Associated prime, Bounded function, Valuation ring, Local ring, Prime ideal

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