Give an example of non split short exact sequence $0\to M\to\Bbb{Z}[X]\to M'\to 0$ of $\Bbb{Z}$-modules.

Question

Give an example of $\Bbb{Z}$-modules $M,M'$ such that $0\to M\to\Bbb{Z}[X]\to M'\to0$ is

• (a) split short exact,
• (b) non split short exact

For (a), take $M=\langle X\rangle=\{Xf(X):\ f(X)\in\Bbb{Z}[X]\}$ and $M'=\Bbb{Z}$ and $0\to \langle X\rangle \hookrightarrow\Bbb{Z}[X]\to \Bbb{Z}[X]/\langle X\rangle\cong \Bbb{Z}\to 0$ is split short exact.

But I am unable to produce an example for (b). Can anyone give me a hint or wayout? Thanks for your help in advance.

Answer 1

Take $M=\mathbb{Z}[X]$ and $M' = \mathbb{Z}/2\mathbb{Z}[X]$, with the map $M \to \mathbb{Z}[X]$ being multiplication by $2$. The sequence can't be split, since any map from $M'$ to $\mathbb{Z}[X]$ is zero (why?).