⊕-supplemented modules relative to an ideal

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.