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Rings whose modules are direct sums of extending modules. (English) Zbl 1176.16006
In the main result of the paper the author shows for a ring \(R\) that every right \(R\)-module is a direct sum of extending modules if and only if \(R\) has finite type and right colocal type. Moreover, in this case \(R\) is Artinian and right serial, and every right \(R\)-module is a direct sum of uniform modules (Theorem 1). Two interesting corollaries can be deduced from this theorem: A ring \(R\) is of right invariant module type if and only if every right \(R\)-module is a direct sum of quasi-injective modules (Corollary 1); A ring \(R\) is Artinian serial if and only if every right \(R\)-module is a direct sum of extending modules and every left \(R\)-module is a direct sum of extending modules.

MSC:
16D70 Structure and classification for modules, bimodules and ideals (except as in 16Gxx), direct sum decomposition and cancellation in associative algebras)
16P20 Artinian rings and modules (associative rings and algebras)
16G60 Representation type (finite, tame, wild, etc.) of associative algebras
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