Let $M$ be a module with $M_1$ and $M_2$ submodules such that their sum (not necessarily a direct sum) is $M$. Is it true in full generality that $\text{Ass}(M) = \text{Ass}(M_1) \cup \text{Ass}(M_2)$? If so prove, if not, provide a counterexample.
I believe the statement is false. The fact that $\text{Ass}(M_i) \subset M$ is obvious, and I know it holds for a direct sum, however I was hoping someone could clarify with a counter example. Thanks is advance.