If $A$ is finitely generated then $C$ is necessary trivial by the fundamental theorem of finitely generated abelian groups. In the case in which $A$ is not finitely generated, I like in particular the following counterexample (which you can find in the Isaac`s book ``Algebra. A graduate course"):  One can see that $({\mathbb R},+)$ is isomorphic to the direct sum of two copy of itself but, of course, $({\mathbb R},+)$ is not trivial...
Anyway, this is not a research level question!