On generalized q.f.d. modules

Mohammad Saleh; S. K. Jain; Sergio R. López-Permouth

Archivum Mathematicum (2005)

  • Volume: 041, Issue: 3, page 243-251
  • ISSN: 0044-8753

Abstract

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A right R -module M is called a generalized q.f.d. module if every M-singular quotient has finitely generated socle. In this note we give several characterizations to this class of modules by means of weak injectivity, tightness, and weak tightness that generalizes the results in [sanh1], Theorem 3. In particular, it is shown that a module M is g.q.f.d. iff every direct sum of M -singular M -injective modules in σ [ M ] is weakly injective iff every direct sum of M -singular weakly tight is weakly tight iff every direct sum of the injective hulls of M -singular simples is weakly R -tight.

How to cite

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Saleh, Mohammad, Jain, S. K., and López-Permouth, Sergio R.. "On generalized q.f.d. modules." Archivum Mathematicum 041.3 (2005): 243-251. <http://eudml.org/doc/249512>.

@article{Saleh2005,
abstract = {A right $R$-module $M$ is called a generalized q.f.d. module if every M-singular quotient has finitely generated socle. In this note we give several characterizations to this class of modules by means of weak injectivity, tightness, and weak tightness that generalizes the results in [sanh1], Theorem 3. In particular, it is shown that a module $M$ is g.q.f.d. iff every direct sum of $M$-singular $M$-injective modules in $\{\sigma [M]\}$ is weakly injective iff every direct sum of $M$-singular weakly tight is weakly tight iff every direct sum of the injective hulls of $M$-singular simples is weakly $R$-tight.},
author = {Saleh, Mohammad, Jain, S. K., López-Permouth, Sergio R.},
journal = {Archivum Mathematicum},
keywords = {tight; weakly tight; weakly injective; q.f.d.; generalized q.f.d. modules; generalized weakly semisimple; generalized weakly semisimple modules; weakly tight modules; weakly injective modules; locally q.f.d. modules; finitely generated submodules; injective hulls; direct sums; simple modules},
language = {eng},
number = {3},
pages = {243-251},
publisher = {Department of Mathematics, Faculty of Science of Masaryk University, Brno},
title = {On generalized q.f.d. modules},
url = {http://eudml.org/doc/249512},
volume = {041},
year = {2005},
}

TY - JOUR
AU - Saleh, Mohammad
AU - Jain, S. K.
AU - López-Permouth, Sergio R.
TI - On generalized q.f.d. modules
JO - Archivum Mathematicum
PY - 2005
PB - Department of Mathematics, Faculty of Science of Masaryk University, Brno
VL - 041
IS - 3
SP - 243
EP - 251
AB - A right $R$-module $M$ is called a generalized q.f.d. module if every M-singular quotient has finitely generated socle. In this note we give several characterizations to this class of modules by means of weak injectivity, tightness, and weak tightness that generalizes the results in [sanh1], Theorem 3. In particular, it is shown that a module $M$ is g.q.f.d. iff every direct sum of $M$-singular $M$-injective modules in ${\sigma [M]}$ is weakly injective iff every direct sum of $M$-singular weakly tight is weakly tight iff every direct sum of the injective hulls of $M$-singular simples is weakly $R$-tight.
LA - eng
KW - tight; weakly tight; weakly injective; q.f.d.; generalized q.f.d. modules; generalized weakly semisimple; generalized weakly semisimple modules; weakly tight modules; weakly injective modules; locally q.f.d. modules; finitely generated submodules; injective hulls; direct sums; simple modules
UR - http://eudml.org/doc/249512
ER -

References

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