On weakly projective and weakly injective modules

Mohammad Saleh

Commentationes Mathematicae Universitatis Carolinae (2004)

  • Volume: 45, Issue: 3, page 389-402
  • ISSN: 0010-2628

Abstract

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The purpose of this paper is to further the study of weakly injective and weakly projective modules as a generalization of injective and projective modules. For a locally q.f.d. module M , there exists a module K ∈ σ [ M ] such that K ⊕ N is weakly injective in σ [ M ] , for any N ∈ σ [ M ] . Similarly, if M is projective and right perfect in σ [ M ] , then there exists a module K ∈ σ [ M ] such that K ⊕ N is weakly projective in σ [ M ] , for any N ∈ σ [ M ] . Consequently, over a right perfect ring every module is a direct summand of a weakly projective module. For some classes ℳ of modules in σ [ M ] , we study when direct sums of modules from ℳ satisfy property ℙ in σ [ M ] . In particular, we get characterizations of locally countably thick modules, a generalization of locally q.f.d. modules.

How to cite

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Saleh, Mohammad. "On weakly projective and weakly injective modules." Commentationes Mathematicae Universitatis Carolinae 45.3 (2004): 389-402. <http://eudml.org/doc/249362>.

@article{Saleh2004,
abstract = {The purpose of this paper is to further the study of weakly injective and weakly projective modules as a generalization of injective and projective modules. For a locally q.f.d. module $M$, there exists a module $K\in \sigma [M]$ such that $K\oplus N$ is weakly injective in $\sigma [M]$, for any $N\in \sigma [M]$. Similarly, if $M$ is projective and right perfect in $\sigma [M]$, then there exists a module $K\in \sigma [M]$ such that $K\oplus N$ is weakly projective in $\sigma [M]$, for any $N\in \sigma [M]$. Consequently, over a right perfect ring every module is a direct summand of a weakly projective module. For some classes $\mathcal \{M\}$ of modules in $\sigma [M]$, we study when direct sums of modules from $\mathcal \{M\}$ satisfy property $\mathbb \{P\}$ in $\sigma [M]$. In particular, we get characterizations of locally countably thick modules, a generalization of locally q.f.d. modules.},
author = {Saleh, Mohammad},
journal = {Commentationes Mathematicae Universitatis Carolinae},
keywords = {tight; weakly tight; weakly injective; weakly projective; countably thick; locally q.f.d.; weakly semisimple; weakly tight modules; weakly injective modules; weakly projective modules; countably thick modules; locally qfd modules; weakly semisimple modules; injective hulls; projective covers},
language = {eng},
number = {3},
pages = {389-402},
publisher = {Charles University in Prague, Faculty of Mathematics and Physics},
title = {On weakly projective and weakly injective modules},
url = {http://eudml.org/doc/249362},
volume = {45},
year = {2004},
}

TY - JOUR
AU - Saleh, Mohammad
TI - On weakly projective and weakly injective modules
JO - Commentationes Mathematicae Universitatis Carolinae
PY - 2004
PB - Charles University in Prague, Faculty of Mathematics and Physics
VL - 45
IS - 3
SP - 389
EP - 402
AB - The purpose of this paper is to further the study of weakly injective and weakly projective modules as a generalization of injective and projective modules. For a locally q.f.d. module $M$, there exists a module $K\in \sigma [M]$ such that $K\oplus N$ is weakly injective in $\sigma [M]$, for any $N\in \sigma [M]$. Similarly, if $M$ is projective and right perfect in $\sigma [M]$, then there exists a module $K\in \sigma [M]$ such that $K\oplus N$ is weakly projective in $\sigma [M]$, for any $N\in \sigma [M]$. Consequently, over a right perfect ring every module is a direct summand of a weakly projective module. For some classes $\mathcal {M}$ of modules in $\sigma [M]$, we study when direct sums of modules from $\mathcal {M}$ satisfy property $\mathbb {P}$ in $\sigma [M]$. In particular, we get characterizations of locally countably thick modules, a generalization of locally q.f.d. modules.
LA - eng
KW - tight; weakly tight; weakly injective; weakly projective; countably thick; locally q.f.d.; weakly semisimple; weakly tight modules; weakly injective modules; weakly projective modules; countably thick modules; locally qfd modules; weakly semisimple modules; injective hulls; projective covers
UR - http://eudml.org/doc/249362
ER -

References

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