Bulletin of the
Korean Mathematical Society
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ISSN(Print) 1015-8634 ISSN(Online) 2234-3016

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Bull. Korean Math. Soc. 2022; 59(2): 453-468

Published online March 31, 2022 https://doi.org/10.4134/BKMS.b210349

Copyright © The Korean Mathematical Society.

Goldie extending property on the class of $z$-closed submodules

Adnan Tercan, Ramazan Ya\c{s}ar, Canan Celep Y\"ucel

Hacettepe University; Hacettepe University; Faculty of Arts and Sciences

Abstract

In this article, we define a module $M$ to be $G^{\, z}$-extending if and only if for each $z$-closed submodule $X$ of $M$ there exists a direct summand $D$ of $M$ such that $X\cap D$ is essential in both $X$ and $D$. We investigate structural properties of $G^{\, z}$-extending modules and locate the implications between the other extending properties. We deal with decomposition theory as well as ring and module extensions for $G^{\, z}$-extending modules. We obtain that if a ring is right $G^{\, z}$-extending, then so is its essential overring. Also it is shown that the $G^{\, z}$-extending property is inherited by its rational hull. Furthermore it is provided some applications including matrix rings over a right $G^{\, z}$-extending ring.

Keywords: Complement, extending module, $z$-closed, $CLS$-module, Goldie extending module, rational Hull

MSC numbers: Primary 16D50, 16D80; Secondary 16D40, 16D70

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