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.

Let R be a ring and M be an R-module. M is called ⊕ss-supplemented if every submodule of M has a ss-supplement that is a direct summand of M. In this paper, the basic properties and characterizations of ⊕ss-supplemented modules are provided. In particular, it is shown that (1) if a module M is ⊕ss-supplemented, then Rad(M) is semisimple and Soc(M) ⊴ M, (2) every direct sum of ss-lifting modules is ⊕ss-supplemented; (3) a commutative ring R is an artinian serial ring with semisimple radical if and only if every left R-module is ⊕ss-supplemented. MSC Classification: 16D10 , 16D60 , 16D99

Let R be an associative ring with identity. We introduce the notion of semi-τ-supplemented modules, which is adapted from srs-modules, for a preradical τ on R-Mod. We provide basic properties of these modules. In particular, we study the objects of R-Mod for τ = Rad. We show that the class of semi-τ-supplemented modules is closed under finite sums and factor modules. We prove that, for an idempotent preradical τ on R-Mod, a module M is semi-τ-supplemented if and only if it is τ-supplemented. For τ = Rad, over a local ring every left module is semi-Rad-supplemented. We also prove that a commutative semilocal ring whose semi-Rad-supplemented modules are a direct sum of w-local left modules is an artinian principal ideal ring.

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