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Dual-ADS, ADS$^\#$ and ADS* Modules
Abyzov Adel Nailevich
,
Bui Tien Dat
,
Truong Cong Quynh
Abstract
A right $R$-module $M$ is said to be dual-ADS if for every decomposition $M=A\oplus B$ then $A$ and $B$ are mutually projective. The class of ADS*-modules contains the class of dual-ADS modules. In this article, we study several properties of these modules. It is shown that a module $M$ is dual-ADS if and only if for any direct summand $S$ and $T^\prime \le M$ with $T^\prime +S$ a direct summand of $M$, then $T^\prime$ contains a direct complement of $S$ in $T^\prime +S$. A generalization of dual-ADS modules is considered, namely, ADS$^\#$-modules. It is shown that a module $M$ is ADS$^\#$ if and only if for any direct summand $S$ of $M$, and any weak supplement $T^\prime$ of $S$ in $T^\prime +S$ such that $T^\prime +S$ is a direct summand of $M$, then $T^\prime$ contains a direct complement of $S$ in $T^\prime +S$.