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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?

  • Maybe you and your classmates can pool your questions instead of duplicating? I'm noticing this was the third version of the question in the span of two days. – rschwieb Nov 18 '19 at 15:29

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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 $\mathbb{Z}$.

Slup
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