A homework problem asked to find a short exact sequence of abelian groups $$0 \rightarrow A \longrightarrow B \longrightarrow C \rightarrow 0$$ such that $B \cong A \oplus C$ although the sequence does not split. My solution to this is the sequence $$0 \rightarrow \mathbb{Z} \overset{i}{\longrightarrow} \mathbb{Z} \oplus (\mathbb{Z}/2\mathbb{Z})^{\mathbb{N}} \overset{p}{\longrightarrow} (\mathbb{Z}/2\mathbb{Z})^{\mathbb{N}} \rightarrow 0$$ with $i(x)=(2x,0,0,\dotsc)$ and $p(x,y_1,y_2,\dotsc)=(x+2\mathbb{Z},y_1,y_2,\dotsc)$.
My new questions:
- Is there an example with finite/finitely generated abelian groups?
- If the answer to$(1)$is negative, will it help to pass to general $R$-modules for some ring $R$?