A module M is called ss-supplemented if every submodule U of M has a supplement V in M such that U \ V is semisimple. It is shown that a …nitely generated module M is ss-supplemented i¤ it is supplemented and Rad(M) Soc(M). A module M is called strongly local if it is local and Rad(M) is semisimple. Any direct sum of strongly local modules is sssupplemented and coatomic. A ring R is semiperfect and Rad(R) Soc(R R) i¤ every left R-module is (amply) ss-supplemented i¤ R R is a …nite sum of strongly local submodules.
Zöschinger studied modules whose radicals have supplements and called these modules radical supplemented. Motivated by this, we call a module strongly radical supplemented (briefly srs) if every submodule containing the radical has a supplement. We prove that every (finitely generated) left module is an srs-module if and only if the ring is left (semi)perfect. Over a local Dedekind domain, srs-modules and radical supplemented modules coincide. Over a nonlocal Dedekind domain, an srs-module is the sum of its torsion submodule and the radical submodule.
In this paper we provide various properties of Rad-⊕-supplemented modules. In particular, we prove that a projective module M is Rad-⊕-supplemented if and only if M is ⊕-supplemented, and then we show that a commutative ring R is an artinian serial ring if and only if every left R-module is Rad-⊕-supplemented. Moreover, every left R-module has the property (P * ) if and only if R is an artinian serial ring and J 2 = 0, where J is the Jacobson radical of R. Finally, we show that every Rad-supplemented module is Rad-⊕-supplemented over dedekind domains.
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