Rings whose modules are direct sums of extending modules

Er, Noyan

doi:10.1090/S0002-9939-09-09807-4

Rings whose modules are direct sums of extending modules
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Proc. Amer. Math. Soc. 137 (2009), 2265-2271 Request permission

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.

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