# Bulletin of theKorean Mathematical SocietyBKMS

ISSN(Print) 1015-8634 ISSN(Online) 2234-3016

## Article

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Bull. Korean Math. Soc. 2021; 58(6): 1563-1567

Published online November 30, 2021 https://doi.org/10.4134/BKMS.b210101

## A decomposition theorem for Utumi and dual-Utumi modules

Yasser Ibrahim, Mohamed Yousif

Taibah University; The Ohio State University

### Abstract

We show that if $M$ is a Utumi module, in particular if $M$ is quasi-continuous, then $M =Q \oplus K$, where $Q$ is quasi-injective that is both a square-full as well as a dual-square-full module, $K$ is a square-free module, and $Q$ \& $K$ are orthogonal. Dually, we also show that if $M$ is a dual-Utumi module whose local summands are summands, in particular if $M$ is quasi-discrete, then $M =P \oplus K$ where $P$ is quasi-projective that is both a square-full as well as a dual-square-full module, $K$ is a dual-square-free module, and $P$ \& $K$ are factor-orthogonal.

Keywords: Utumi and dual Utumi modules, quasi-injective and quasi-projective modules, square-free and dual-square-free modules, discrete and quasi-discrete modules, $D 3$-modules and $D 4$-modules

MSC numbers: Primary 16D40, 16D50, 16D60; Secondary 16L30, 16P20, 16P60