Journal of Commutative Algebra

Indecomposable injective modules of finite Malcev rank over local commutative rings

François Couchot

Full-text: Open access

Abstract

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

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

Dates
First available in Project Euclid: 31 January 2014

Permanent link to this document
https://projecteuclid.org/euclid.jca/1391192653

Digital Object Identifier
doi:10.1216/JCA-2013-5-4-481

Mathematical Reviews number (MathSciNet)
MR3161743

Zentralblatt MATH identifier
1291.13037

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

Keywords
Chain ring valuation domain polyserial module indecomposable injective module Goldie dimension

Citation

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. https://projecteuclid.org/euclid.jca/1391192653


Export citation

References

  • F. Couchot, Les modules artiniens et leurs enveloppes quasi-injectives, on Séminaire d'algèbre Paul Dubreuil et Marie-Paule Malliavin, Lect. Notes Math.867 (1981), 380-395.
  • –––, Injective modules and fp-injective modules over valuation rings, J. Algebra 267 (2003), 359-376.
  • –––, Local rings of bounded module type are almost maximal valuation rings, Comm. Algebra 33 (2005), 2851-2855.
  • –––, Localization of injective modules over valuations rings, Proc. Amer. Math. Soc. 134 (2006), 1013-1017.
  • –––, Pure-injective hulls of modules over valuation rings, J. Pure Appl. Algebra, 207 (2006), 63-76.
  • –––, Flat modules over valuation rings, J. Pure Appl. Alg. 211 (2007), 235-247.
  • –––, Valuation domains with a maximal immediate extension of finite rank, J. Algebra 323 (2010, 32-41.
  • A. Facchini and P. Zanardo, Discrete valuation domains and ranks of their maximal extensions, Rend. Sem. Mat. Univ. Padova 75 1986), 143-156.
  • L. Fuchs and L. Salce, Modules over valuation domains, Lect. Notes Pure Appl. Math. 97 (1985), Marcel Dekker, New York.
  • –––, Modules over non-Noetherian domains, Math. Surv. Mono. 84 (2001), American Mathematical Society, Providence, RI.
  • D.T. Gill, Almost maximal valuation rings, J. Lond. Math. Soc. 4 (1971), 140-146.
  • G.B. Klatt and L.S. Levy, Pre-self injectives rings, Trans. Amer. Math. Soc. 137 (1969), 407-419.
  • M.P. Malliavin-Brameret, Largeur d'anneaux et de modules, Bull. Soc. Math. France 8, (1966).
  • E. Matlis, Injective modules over Noetherian rings, Pacific J. Math. 8 (1958), 511-528.
  • –––, Injective modules over Prüfer rings, Nagoya Math. J. 15 (1959), 57-69.
  • H. Tsang. Gauss's lemma, Ph.D. thesis, University of Chicago, 1965.
  • P. Vámos, Classical rings, J. Alg. 34 (1975), 114-129.
  • P. Vámos, Decomposition problems for modules over valuation domains, J. Lond. Math. Soc. 41 (1990), 10-26.
  • R.B. Warfield, Purity and algebraic compactness for modules, Pacific J. Math. 28 (1969), 689-719.
  • R. Wiegand and S. Wiegand, Finitely generated modules over Bezout rings, Pacific J. Math. 58 (1975), 655-664. \noindentstyle