2017
Том 69
№ 9

All Issues

Co-coatomically supplemented modules

Alizade R., Güngör S.


Abstract

It is shown that if a submodule $N$ of $M$ is co-coatomically supplemented and $M/N$ has no maximal submodule, then $M$ is a co-coatomically supplemented module. If a module $M$ is co-coatomically supplemented, then every finitely $M$-generated module is a co-coatomically supplemented module. Every left $R$-module is co-coatomically supplemented if and only if the ring $R$ is left perfect. Over a discrete valuation ring, a module $M$ is co-coatomically supplemented if and only if the basic submodule of $M$ is coatomic. Over a nonlocal Dedekind domain, if the torsion part $T(M)$ of a reduced module $M$ has a weak supplement in $M$, then $M$ is co-coatomically supplemented if and only if $M/T (M)$ is divisible and $TP (M)$ is bounded for each maximal ideal $P$. Over a nonlocal Dedekind domain, if a reduced module $M$ is co-coatomically amply supplemented, then $M/T (M)$ is divisible and $TP (M)$ is bounded for each maximal ideal $P$. Conversely, if $M/T (M)$ is divisible and $TP (M)$ is bounded for each maximal ideal $P$, then $M$ is a co-coatomically supplemented module.

Citation Example: Alizade R., Güngör S. Co-coatomically supplemented modules // Ukr. Mat. Zh. - 2017. - 69, № 7. - pp. 867-876.