Co-coatomically supplemented modules

• Datum: 2017
• Hauptverfasser: Alizade, R., Güngör, S., Алізаде, Р., Гюнгер, С.
• Format: Artikel
• Sprache: Englisch
• Veröffentlicht: Institute of Mathematics, NAS of Ukraine 2017
• Online Zugang: https://umj.imath.kiev.ua/index.php/umj/article/view/1742
• Назва журналу: Ukrains’kyi Matematychnyi Zhurnal

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.