What is a simple example of two abelian groups $A,B$ which are isomorphic to direct summands of each other (that is, $A \cong B + C$ and $B \cong A + D$ for some abelian groups $C,D$), but which are not isomorphic?
Currently I only know the following: There are abelian groups $A$ with $A \cong A^3$ but $A \not\cong A^2$. Then we may take $B=A^2$. However, the construction of these $A$ is quite complicated, as far as I know.