On commutative rings whose maximal ideals are idempotent

Farid Kourki; Rachid Tribak

Commentationes Mathematicae Universitatis Carolinae (2019)

  • Volume: 60, Issue: 3, page 313-322
  • ISSN: 0010-2628

Abstract

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We prove that for a commutative ring R , every noetherian (artinian) R -module is quasi-injective if and only if every noetherian (artinian) R -module is quasi-projective if and only if the class of noetherian (artinian) R -modules is socle-fine if and only if the class of noetherian (artinian) R -modules is radical-fine if and only if every maximal ideal of R is idempotent.

How to cite

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Kourki, Farid, and Tribak, Rachid. "On commutative rings whose maximal ideals are idempotent." Commentationes Mathematicae Universitatis Carolinae 60.3 (2019): 313-322. <http://eudml.org/doc/294355>.

@article{Kourki2019,
abstract = {We prove that for a commutative ring $R$, every noetherian (artinian) $R$-module is quasi-injective if and only if every noetherian (artinian) $R$-module is quasi-projective if and only if the class of noetherian (artinian) $R$-modules is socle-fine if and only if the class of noetherian (artinian) $R$-modules is radical-fine if and only if every maximal ideal of $R$ is idempotent.},
author = {Kourki, Farid, Tribak, Rachid},
journal = {Commentationes Mathematicae Universitatis Carolinae},
keywords = {artinian module; modules of finite length; noetherian module; quasi-injective module; quasi-projective module; radical-fine class of modules; socle-fine class of modules},
language = {eng},
number = {3},
pages = {313-322},
publisher = {Charles University in Prague, Faculty of Mathematics and Physics},
title = {On commutative rings whose maximal ideals are idempotent},
url = {http://eudml.org/doc/294355},
volume = {60},
year = {2019},
}

TY - JOUR
AU - Kourki, Farid
AU - Tribak, Rachid
TI - On commutative rings whose maximal ideals are idempotent
JO - Commentationes Mathematicae Universitatis Carolinae
PY - 2019
PB - Charles University in Prague, Faculty of Mathematics and Physics
VL - 60
IS - 3
SP - 313
EP - 322
AB - We prove that for a commutative ring $R$, every noetherian (artinian) $R$-module is quasi-injective if and only if every noetherian (artinian) $R$-module is quasi-projective if and only if the class of noetherian (artinian) $R$-modules is socle-fine if and only if the class of noetherian (artinian) $R$-modules is radical-fine if and only if every maximal ideal of $R$ is idempotent.
LA - eng
KW - artinian module; modules of finite length; noetherian module; quasi-injective module; quasi-projective module; radical-fine class of modules; socle-fine class of modules
UR - http://eudml.org/doc/294355
ER -

References

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