SS-supplemented modules

Abstract

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.

Strongly ⨁-Locally Artinian Supplemented Modules

“…It is clear that the inclusions 𝑆𝑜𝑐 𝑠 (𝑀) ⊆ 𝑅𝑎𝑑(𝑀) and 𝑆𝑜𝑐 𝑠 (𝑀) ⊆ 𝑆𝑜𝑐(𝑀) are hold. In (Kaynar et al, 2020) In section 3, as a strong notion of strongly ⨁-supplemented modules, we define strongly ⨁-locally artinian supplemented modules and provide the basic properties of strongly ⨁-locally artinian supplemented modules. Especially, we show that every direct summand of a strongly ⨁-locally artinian supplemented module is strongly ⨁-locally artinian supplemented in Proposition 3.4.…”

δss -supplemented modules and rings

“…In [6], the authors define ss-supplemented modules as a proper generalization of semisimple modules. A module M is said to be ss-supplemented if every submodule U of M has a supplement V in M such that U ∩V is semisimple.…”

“…They give in the same paper the structure of ss-supplemented modules. In particular, it is shown in [6,Theorem 41] that a ring R is semiperfect and Rad(R) ⊆ Soc( R R) if and only if every left R-module is ss-supplemented if and only if R R is the finite sum of strongly local submodules. Here a module M is called strongly local if it is local and the radical is semisimple ( [6]).…”

“…As we have mentioned in the introduction, a module M is strongly local if it is local and Rad(M ) is semisimple ( [6]). Note that every simple module is strongly local.…”

“…Modifying of[6, Lemma 3] we characterize δ ss -supplement submodules of a module M . Note that we shall freely use the next lemma without reference in this paper.Lemma 3.3.…”