# Definition of semisimple modules

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"?

• 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. – Crostul Nov 20 '16 at 21:48
• Thank you, that is a nice example – adiselann Nov 20 '16 at 22:18

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.