Journal of Commutative Algebra

Indecomposable injective modules of finite Malcev rank over local commutative rings

François Couchot

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It is proven that each indecomposable injective module over a valuation domain $R$ is polyserial if and only if each maximal immediate extension $\widehat{R}$ of $R$ is of finite rank over the completion $\widetilde{R}$ of $R$ in the $R$-topology. In this case, for each indecomposable injective module $E$, the following invariants are finite and equal: its Malcev rank, its Fleischer rank and its dual Goldie dimension. Similar results are obtained for chain rings satisfying some additional properties. It is also shown that each indecomposable injective module over local Noetherian rings of Krull dimension one has finite Malcev rank. The preservation of the finiteness of Goldie dimension by localization is investigated too.

Article information

J. Commut. Algebra, Volume 5, Number 4 (2013), 481-505.

First available in Project Euclid: 31 January 2014

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Zentralblatt MATH identifier

Primary: 13F30: Valuation rings [See also 13A18] 13C11: Injective and flat modules and ideals 13E05: Noetherian rings and modules

Chain ring valuation domain polyserial module indecomposable injective module Goldie dimension


Couchot, François. Indecomposable injective modules of finite Malcev rank over local commutative rings. J. Commut. Algebra 5 (2013), no. 4, 481--505. doi:10.1216/JCA-2013-5-4-481.

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