Question

If every submodule is a direct summand, then it is semisimple

Answer 1

Explanation:

Next, we show that MM is semisimple. By Zorn’s Lemma, there is a family (Sk)k∈K(Sk)k∈K of simple submodules of M maximal with the property that they generate their direct sum D=⨁k∈KSkD=⨁k∈KSk .

By hypothesis, M=D⊕EM=D⊕E for some submodule EE. If E={0}E={0}, we are done. Otherwise, 

Answer 2

Answer:

A module M is called t-semisimple if every submodule N contains a direct summand K of M such that K is t-essential in N. T-semisimple modules are Morita invariant and they form a strict subclass of t-extending modules.