$$H$-supplemented modules with respect to a preradical$

$H$-supplemented modules with respect to a preradical

Let $M$ be a right $R$-module and $\tau$ a preradical. We call $M$ $\tau$-$H$-supplemented if for every submodule $A$ of $M$ there exists a direct summand $D$ of $M$ such that $(A + D)/D \subseteq \tau(M/D)$ and $(A + D)/A \subseteq \tau(M/A)$. Let $\tau$ be a cohereditary...

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Date:	2018
Main Authors:	Talebi, Yahya, Hamzekolaei, A. R. Moniri, Tutuncu, Derya Keskin
Format:	Article
Language:	English
Published:	Lugansk National Taras Shevchenko University 2018
Subjects:	$H$-supplemented module $\tau$-$H$-supplemented module $\tau$-lifting module 16S90 16D10 16D70 16D99
Online Access:	https://admjournal.luguniv.edu.ua/index.php/adm/article/view/675
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Journal Title:	Algebra and Discrete Mathematics

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Algebra and Discrete Mathematics

id	oai:ojs.admjournal.luguniv.edu.ua:article-675
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spelling	oai:ojs.admjournal.luguniv.edu.ua:article-6752018-04-04T09:28:39Z $H$-supplemented modules with respect to a preradical Talebi, Yahya Hamzekolaei, A. R. Moniri Tutuncu, Derya Keskin $H$-supplemented module, $\tau$-$H$-supplemented module, $\tau$-lifting module 16S90, 16D10, 16D70, 16D99 Let $M$ be a right $R$-module and $\tau$ a preradical. We call $M$ $\tau$-$H$-supplemented if for every submodule $A$ of $M$ there exists a direct summand $D$ of $M$ such that $(A + D)/D \subseteq \tau(M/D)$ and $(A + D)/A \subseteq \tau(M/A)$. Let $\tau$ be a cohereditary preradical. Firstly, for a duo module $M = M_{1} \oplus M_{2}$ we prove that $M$ is $\tau$-$H$-supplemented if and only if $M_{1}$ and $M_{2}$ are $\tau$-$H$-supplemented. Secondly, let $M=\oplus_{i=1}^nM_i$ be a $\tau$-supplemented module. Assume that $M_i$ is $\tau$-$M_j$-projective for all $j > i$. If each $M_i$ is $\tau$-$H$-supplemented, then $M$ is $\tau$-$H$-supplemented. We also investigate the relations between $\tau$-$H$-supplemented modules and $\tau$-($\oplus$-)supplemented modules. Lugansk National Taras Shevchenko University 2018-04-04 Article Article Peer-reviewed Article application/pdf https://admjournal.luguniv.edu.ua/index.php/adm/article/view/675 Algebra and Discrete Mathematics; Vol 12, No 1 (2011) 2415-721X 1726-3255 en https://admjournal.luguniv.edu.ua/index.php/adm/article/view/675/209 Copyright (c) 2018 Algebra and Discrete Mathematics
institution	Algebra and Discrete Mathematics
baseUrl_str
datestamp_date	2018-04-04T09:28:39Z
collection	OJS
language	English
topic	$H$-supplemented module $\tau$-$H$-supplemented module $\tau$-lifting module 16S90 16D10 16D70 16D99
spellingShingle	$H$-supplemented module $\tau$-$H$-supplemented module $\tau$-lifting module 16S90 16D10 16D70 16D99 Talebi, Yahya Hamzekolaei, A. R. Moniri Tutuncu, Derya Keskin $H$-supplemented modules with respect to a preradical
topic_facet	$H$-supplemented module $\tau$-$H$-supplemented module $\tau$-lifting module 16S90 16D10 16D70 16D99
format	Article
author	Talebi, Yahya Hamzekolaei, A. R. Moniri Tutuncu, Derya Keskin
author_facet	Talebi, Yahya Hamzekolaei, A. R. Moniri Tutuncu, Derya Keskin
author_sort	Talebi, Yahya
title	$H$-supplemented modules with respect to a preradical
title_short	$H$-supplemented modules with respect to a preradical
title_full	$H$-supplemented modules with respect to a preradical
title_fullStr	$H$-supplemented modules with respect to a preradical
title_full_unstemmed	$H$-supplemented modules with respect to a preradical
title_sort	$h$-supplemented modules with respect to a preradical
description	Let $M$ be a right $R$-module and $\tau$ a preradical. We call $M$ $\tau$-$H$-supplemented if for every submodule $A$ of $M$ there exists a direct summand $D$ of $M$ such that $(A + D)/D \subseteq \tau(M/D)$ and $(A + D)/A \subseteq \tau(M/A)$. Let $\tau$ be a cohereditary preradical. Firstly, for a duo module $M = M_{1} \oplus M_{2}$ we prove that $M$ is $\tau$-$H$-supplemented if and only if $M_{1}$ and $M_{2}$ are $\tau$-$H$-supplemented. Secondly, let $M=\oplus_{i=1}^nM_i$ be a $\tau$-supplemented module. Assume that $M_i$ is $\tau$-$M_j$-projective for all $j > i$. If each $M_i$ is $\tau$-$H$-supplemented, then $M$ is $\tau$-$H$-supplemented. We also investigate the relations between $\tau$-$H$-supplemented modules and $\tau$-($\oplus$-)supplemented modules.
publisher	Lugansk National Taras Shevchenko University
publishDate	2018
url	https://admjournal.luguniv.edu.ua/index.php/adm/article/view/675
work_keys_str_mv	AT talebiyahya hsupplementedmoduleswithrespecttoapreradical AT hamzekolaeiarmoniri hsupplementedmoduleswithrespecttoapreradical AT tutuncuderyakeskin hsupplementedmoduleswithrespecttoapreradical
first_indexed	2025-07-17T10:36:30Z
last_indexed	2025-07-17T10:36:30Z
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\(H\)-supplemented modules with respect to a preradical

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