23
$\begingroup$

A homework problem asked to find a short exact sequence of abelian groups $$0 \rightarrow A \rightarrow B \rightarrow C \rightarrow 0$$ such that $$B \cong A \oplus C$$ although the sequence does not split.

My solution: $$0 \rightarrow \mathbb{Z} \overset{i}{\rightarrow} \mathbb{Z} \oplus (\mathbb{Z}/2\mathbb{Z})^{\mathbb{N}} \overset{p}{\rightarrow} (\mathbb{Z}/2\mathbb{Z})^{\mathbb{N}} \rightarrow 0$$ 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:

  1. Is there an example with finite/finitely generated abelian groups?
  2. If the answer to $(1)$ is negative, will it help to pass to general $R$-modules for some ring $R$?
  • 4
    FYI: A different example, still with infinite groups, is Example 6.35 of Rotman's Advanced Modern Algebra (2nd ed., AMS).2012-04-22

3 Answers 3