# $\oplus$-cofinitely supplemented modules

• Volume: 54, Issue: 4, page 1083-1088
• ISSN: 0011-4642

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## Abstract

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Let $R$ be a ring and $M$ a right $R$-module. $M$ is called $\oplus$-cofinitely supplemented if every submodule $N$ of $M$ with $\frac{M}{N}$ finitely generated has a supplement that is a direct summand of $M$. In this paper various properties of the $\oplus$-cofinitely supplemented modules are given. It is shown that (1) Arbitrary direct sum of $\oplus$-cofinitely supplemented modules is $\oplus$-cofinitely supplemented. (2) A ring $R$ is semiperfect if and only if every free $R$-module is $\oplus$-cofinitely supplemented. In addition, if $M$ has the summand sum property, then $M$ is $\oplus$-cofinitely supplemented iff every maximal submodule has a supplement that is a direct summand of $M$.

## How to cite

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Çalışıcı, H., and Pancar, A.. "$\oplus$-cofinitely supplemented modules." Czechoslovak Mathematical Journal 54.4 (2004): 1083-1088. <http://eudml.org/doc/30923>.

@article{Çalışıcı2004,
abstract = {Let $R$ be a ring and $M$ a right $R$-module. $M$ is called $\oplus$-cofinitely supplemented if every submodule $N$ of $M$ with $\frac\{M\}\{N\}$ finitely generated has a supplement that is a direct summand of $M$. In this paper various properties of the $\oplus$-cofinitely supplemented modules are given. It is shown that (1) Arbitrary direct sum of $\oplus$-cofinitely supplemented modules is $\oplus$-cofinitely supplemented. (2) A ring $R$ is semiperfect if and only if every free $R$-module is $\oplus$-cofinitely supplemented. In addition, if $M$ has the summand sum property, then $M$ is $\oplus$-cofinitely supplemented iff every maximal submodule has a supplement that is a direct summand of $M$.},
author = {Çalışıcı, H., Pancar, A.},
journal = {Czechoslovak Mathematical Journal},
keywords = {cofinite submodule; $\oplus$-cofinitely supplemented module; cofinite submodules; cofinitely supplemented modules; summand sum property; direct summands; semiperfect rings},
language = {eng},
number = {4},
pages = {1083-1088},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {$\oplus$-cofinitely supplemented modules},
url = {http://eudml.org/doc/30923},
volume = {54},
year = {2004},
}

TY - JOUR
AU - Çalışıcı, H.
AU - Pancar, A.
TI - $\oplus$-cofinitely supplemented modules
JO - Czechoslovak Mathematical Journal
PY - 2004
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 54
IS - 4
SP - 1083
EP - 1088
AB - Let $R$ be a ring and $M$ a right $R$-module. $M$ is called $\oplus$-cofinitely supplemented if every submodule $N$ of $M$ with $\frac{M}{N}$ finitely generated has a supplement that is a direct summand of $M$. In this paper various properties of the $\oplus$-cofinitely supplemented modules are given. It is shown that (1) Arbitrary direct sum of $\oplus$-cofinitely supplemented modules is $\oplus$-cofinitely supplemented. (2) A ring $R$ is semiperfect if and only if every free $R$-module is $\oplus$-cofinitely supplemented. In addition, if $M$ has the summand sum property, then $M$ is $\oplus$-cofinitely supplemented iff every maximal submodule has a supplement that is a direct summand of $M$.
LA - eng
KW - cofinite submodule; $\oplus$-cofinitely supplemented module; cofinite submodules; cofinitely supplemented modules; summand sum property; direct summands; semiperfect rings
UR - http://eudml.org/doc/30923
ER -

## References

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5. Continuous and Discrete Modules. London Math. Soc. LNS Vol.  147, Cambridge Univ. Press, Cambridge, 1990. (1990) MR1084376
6. Foundations of Module and Ring Theory, Gordon and Breach, Philadelphia, 1991. (1991) Zbl0746.16001MR1144522
7. 10.1016/0021-8693(74)90109-4, J.  Algebra 29 (1974), 42–56. (1974) MR0340347DOI10.1016/0021-8693(74)90109-4

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