Some results on quasi-t-dual Baer modules

Rachid Tribak; Yahya Talebi; Mehrab Hosseinpour

Commentationes Mathematicae Universitatis Carolinae (2023)

  • Volume: 64, Issue: 4, page 411-427
  • ISSN: 0010-2628

Abstract

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Let R be a ring and let M be an R -module with S = End R ( M ) . Consider the preradical Z ¯ for the category of right R -modules Mod- R introduced by Y. Talebi and N. Vanaja in 2002 and defined by Z ¯ ( M ) = ⋂ { U ≤ M : M / U is small in its injective hull } . The module M is called quasi-t-dual Baer if ∑ ϕ ∈ ℑ ϕ ( Z ¯ 2 ( M ) ) is a direct summand of M for every two-sided ideal ℑ of S , where Z ¯ 2 ( M ) = Z ¯ ( Z ¯ ( M ) ) . In this paper, we show that M is quasi-t-dual Baer if and only if Z ¯ 2 ( M ) is a direct summand of M and Z ¯ 2 ( M ) is a quasi-dual Baer module. It is also shown that any direct summand of a quasi-t-dual Baer module inherits the property. The last part of the paper is devoted to the comparison of the notions of quasi-dual Baer modules and quasi-t-dual Baer modules. Also, right quasi-t-dual Baer rings are investigated.

How to cite

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Tribak, Rachid, Talebi, Yahya, and Hosseinpour, Mehrab. "Some results on quasi-t-dual Baer modules." Commentationes Mathematicae Universitatis Carolinae 64.4 (2023): 411-427. <http://eudml.org/doc/299325>.

@article{Tribak2023,
abstract = {Let $R$ be a ring and let $M$ be an $R$-module with $S=\rm \{End\}_R(M)$. Consider the preradical $\{\hspace\{0.83328pt\}\overline\{\hspace\{-1.94443pt\}Z \hspace\{-0.27771pt\}\}\hspace\{0.83328pt\}\}$ for the category of right $R$-modules Mod-$R$ introduced by Y. Talebi and N. Vanaja in 2002 and defined by $\{\hspace\{0.83328pt\}\overline\{\hspace\{-1.94443pt\}Z \hspace\{-0.27771pt\}\}\hspace\{0.83328pt\}\}(M) = \bigcap \lbrace U\le M\colon M/U$ is small in its injective hull$\rbrace $. The module $M$ is called quasi-t-dual Baer if $\sum _\{\varphi \in \mathfrak \{I\}\} \varphi (\{\{\hspace\{0.83328pt\}\overline\{\hspace\{-1.94443pt\}Z \hspace\{-0.27771pt\}\}\hspace\{0.83328pt\}\}\}^2(M))$ is a direct summand of $M$ for every two-sided ideal $\mathfrak \{I\}$ of $S$, where $\{\{\hspace\{0.83328pt\}\overline\{\hspace\{-1.94443pt\}Z \hspace\{-0.27771pt\}\}\hspace\{0.83328pt\}\}\}^2(M) = \{\{\hspace\{0.83328pt\}\overline\{\hspace\{-1.94443pt\}Z \hspace\{-0.27771pt\}\}\hspace\{0.83328pt\}\}\} (\{\{\hspace\{0.83328pt\}\overline\{\hspace\{-1.94443pt\}Z \hspace\{-0.27771pt\}\}\hspace\{0.83328pt\}\}\}(M))$. In this paper, we show that $M$ is quasi-t-dual Baer if and only if $\{\{\hspace\{0.83328pt\}\overline\{\hspace\{-1.94443pt\}Z \hspace\{-0.27771pt\}\}\hspace\{0.83328pt\}\}\}^2(M) $ is a direct summand of $M$ and $\{\hspace\{0.83328pt\}\overline\{\hspace\{-1.94443pt\}Z \hspace\{-0.27771pt\}\}\hspace\{0.83328pt\}\}^2(M)$ is a quasi-dual Baer module. It is also shown that any direct summand of a quasi-t-dual Baer module inherits the property. The last part of the paper is devoted to the comparison of the notions of quasi-dual Baer modules and quasi-t-dual Baer modules. Also, right quasi-t-dual Baer rings are investigated.},
author = {Tribak, Rachid, Talebi, Yahya, Hosseinpour, Mehrab},
journal = {Commentationes Mathematicae Universitatis Carolinae},
keywords = {fully invariant submodule; quasi-dual Baer module; quasi-dual Baer ring; quasi-t-dual Baer module; quasi-t-dual Baer ring},
language = {eng},
number = {4},
pages = {411-427},
publisher = {Charles University in Prague, Faculty of Mathematics and Physics},
title = {Some results on quasi-t-dual Baer modules},
url = {http://eudml.org/doc/299325},
volume = {64},
year = {2023},
}

TY - JOUR
AU - Tribak, Rachid
AU - Talebi, Yahya
AU - Hosseinpour, Mehrab
TI - Some results on quasi-t-dual Baer modules
JO - Commentationes Mathematicae Universitatis Carolinae
PY - 2023
PB - Charles University in Prague, Faculty of Mathematics and Physics
VL - 64
IS - 4
SP - 411
EP - 427
AB - Let $R$ be a ring and let $M$ be an $R$-module with $S=\rm {End}_R(M)$. Consider the preradical ${\hspace{0.83328pt}\overline{\hspace{-1.94443pt}Z \hspace{-0.27771pt}}\hspace{0.83328pt}}$ for the category of right $R$-modules Mod-$R$ introduced by Y. Talebi and N. Vanaja in 2002 and defined by ${\hspace{0.83328pt}\overline{\hspace{-1.94443pt}Z \hspace{-0.27771pt}}\hspace{0.83328pt}}(M) = \bigcap \lbrace U\le M\colon M/U$ is small in its injective hull$\rbrace $. The module $M$ is called quasi-t-dual Baer if $\sum _{\varphi \in \mathfrak {I}} \varphi ({{\hspace{0.83328pt}\overline{\hspace{-1.94443pt}Z \hspace{-0.27771pt}}\hspace{0.83328pt}}}^2(M))$ is a direct summand of $M$ for every two-sided ideal $\mathfrak {I}$ of $S$, where ${{\hspace{0.83328pt}\overline{\hspace{-1.94443pt}Z \hspace{-0.27771pt}}\hspace{0.83328pt}}}^2(M) = {{\hspace{0.83328pt}\overline{\hspace{-1.94443pt}Z \hspace{-0.27771pt}}\hspace{0.83328pt}}} ({{\hspace{0.83328pt}\overline{\hspace{-1.94443pt}Z \hspace{-0.27771pt}}\hspace{0.83328pt}}}(M))$. In this paper, we show that $M$ is quasi-t-dual Baer if and only if ${{\hspace{0.83328pt}\overline{\hspace{-1.94443pt}Z \hspace{-0.27771pt}}\hspace{0.83328pt}}}^2(M) $ is a direct summand of $M$ and ${\hspace{0.83328pt}\overline{\hspace{-1.94443pt}Z \hspace{-0.27771pt}}\hspace{0.83328pt}}^2(M)$ is a quasi-dual Baer module. It is also shown that any direct summand of a quasi-t-dual Baer module inherits the property. The last part of the paper is devoted to the comparison of the notions of quasi-dual Baer modules and quasi-t-dual Baer modules. Also, right quasi-t-dual Baer rings are investigated.
LA - eng
KW - fully invariant submodule; quasi-dual Baer module; quasi-dual Baer ring; quasi-t-dual Baer module; quasi-t-dual Baer ring
UR - http://eudml.org/doc/299325
ER -

References

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