Rings whose modules are direct sums of extending modules

Abstract

We prove that for a ring $R$, the following are equivalent: (i) Every right $R$-module is a direct sum of extending modules, and (ii) $R$ has finite type and right colocal type (i.e., every indecomposable right $R$-module has simple socle). Thus, in this case, $R$ is two-sided Artinian and right serial, and every right $R$-module is a direct sum of finitely generated uniform modules. This property of a ring is not left-right symmetric. A consequence is the following: $R$ is Artinian serial if and only if every $R$-module is a direct sum of extending modules if and only if $R$ is left serial with every right $R$-module a direct sum of extending modules.