Abstract
We will prove that over a chain domain with Krull dimension each pure projective module decomposes into a direct sum of finitely presented modules.
Highlights
A ring is said to be a chain ring if its right ideals are linearly ordered by inclusion, and the same holds true for its left ideals.✩ The paper was written during two stays of the second author in ECI, Prague
It follows from Warfield that every pure projective module over a chain ring is isomorphic to a direct summand of a direct sum of uniserial modules R/rR
If each module R/rR has a local endomorphism ring, the extended version of the Crawley–Jønsson theorem shows that every pure projective module is isomorphic to a direct sum of finitely presented modules
Summary
A ring is said to be a chain ring if its right ideals are linearly ordered by inclusion, and the same holds true for its left ideals. If each module R/rR has a local endomorphism ring, the extended version of the Crawley–Jønsson theorem (see [5, Theorem 5.2]) shows that every pure projective module is isomorphic to a direct sum of finitely presented modules For example this is the case when R is a commutative chain ring. In this paper we will completely characterize chain domains R such that every pure projective R-module is isomorphic to a direct sum of finitely presented modules. In this paper we will suggest a very abstract version of this construction that covers all known examples of strange pure projective modules over chain rings By evaluating it for various dimensions we will identify the main obstacles to the existence of a perfect ( trivial) decomposition theory of pure projectives. We are indebted to the referee for a very careful reading of the manuscript, which greatly improved the presentation
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