I've seen following definition: an $R$-module $M$ is semisimple if every submodule of $M$ has a complement.
Does anyone have example of a module which is not semisimple in $\mathbb{Z}$, $\mathbb{C}[t]$ and $\mathbb{C}[\mathbb{Z}]$?
I think $\mathbb{Z}$ is the module which is not semisimple in $\mathbb{Z}$. But I couldn't find module which is not semisimple in $\mathbb{C}[t]$ or $\mathbb{C}[\mathbb{Z}]$. Does anyone have an example?
Fact. For every nontrivial submodule $N$ of $\mathbb{Z}$ we have $N\cap 2\mathbb{Z}\neq 0$.
Proof. Indeed, for every nontrivial submodule $N\subseteq \mathbb{Z}$ pick $n\in N\setminus \{0\}$. Then $2n\in 2\mathbb{Z}\cap N$.
In particular, $2\mathbb{Z}\subseteq \mathbb{Z}$ has no complement.
For $\mathbb{C}[t]$ pick $t\cdot \mathbb{C}[t]\subseteq \mathbb{C}[t]$ and use analogical argument as for $\mathbb{Z}$.
It remains to check the result for group algebra $\mathbb{C}[\mathbb{Z}]$. Note that $\mathbb{C}[\mathbb{Z}]\cong \mathbb{C}[t,t^{-1}]$. Pick $(t-1)\cdot \mathbb{C}[t,t^{-1}]\subseteq \mathbb{C}[t,t^{-1}]$ and use analogical argument as for $\mathbb{Z}$.