A note on a generalization of injective modules

Keywords: supplement, mutual supplement, module with the property (ME), left perfect ring
Published online: 2020-12-30


As a proper generalization of injective modules in term of supplements, we say that a module $M$ has the property (ME) if, whenever $M\subseteq N$, $M$ has a supplement $K$ in $N$, where $K$ has a mutual supplement in $N$. In this study, we obtain that $(1)$ a semisimple $R$-module $M$ has the property (E) if and only if $M$ has the property (ME); $(2)$ a semisimple left $R$-module $M$ over a commutative Noetherian ring $R$ has the property (ME) if and only if $M$ is algebraically compact if and only if almost all isotopic components of $M$ are zero; $(3)$ a module $M$ over a von Neumann regular ring has the property (ME) if and only if it is injective; $(4)$ a principal ideal domain $R$ is left perfect if every free left $R$-module has the property (ME)

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