⊕-supplemented modules relative to an ideal

Year 2016, Volume: 45 Issue: 1, 107 - 120, 01.02.2016

Rachid Tribak , Yahya Talebi , Ali Reza Moniri Hamzekolaee , Samira Asgari

Abstract

Let $I$ be an ideal of a ring $R$ and let $M$ be a left
$R$-module. A submodule $L$ of $M$ is said to be $\delta$-small in $M$ provided
$M \neq L + X$ for any proper submodule $X$ of $M$ with $M/X$ singular. An
$R$-module $M$ is called $I-\bigoplus$-supplemented if for every submodule $N$ of $M$, there
exists a direct summand $K$ of $M$ such that $M = N + K$, $N \cap K \subseteq
IK$ and $N \cap K$ is $\delta$-small in $K$. In this paper, we investigate some
properties of $I-\bigoplus$-supplemented modules. We also compare $I-\bigoplus$-supplemented
modules with $\bigoplus$-supplemented modules. The structure of $I-\bigoplus$-supplemented
modules and $\bigoplus-\delta$-supplemented modules over a Dedekind domain is
completely determined.

Keywords

$\delta$-small submodules, $\bigoplus-\delta$-supplemented modules, $\bigoplus$-supplemented modules, $I-\bigoplus$-supplemented modules

References

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