GOLDIE EXTENDING PROPERTY ON THE CLASS OF $z$-CLOSED SUBMODULES
Bull. Korean Math. Soc.
Published online November 4, 2021
Adnan Tercan, Ramazan Yaşar, and Canan Celep Yücel
Hacettepe University; Hacettepe University; Pamukkale University
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 : 16D50, 16D80;16D40, 16D70
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