M is a semisimple module iff every submodule of M is a direct summand (here is the definition of semisimple modules, and this is the property 3), but this property can be replaced with "every submodule of M is $\textit{isomorphic}$ to a direct summand of M"?
2026-04-17 08:20:47.1776414047
Definition of semisimple modules
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Of course not. For example, every submodule of $\Bbb{Z}$ is isomorphic either to $\Bbb{Z}$ or $0$, which are direct summands of $\Bbb{Z}$, however $\Bbb{Z}$ is not a semisimple $\Bbb{Z}$-module.