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I'm looking for a module $E$ of a commutative ring $A$ with two submodules $E_1$ and $E_2$ such that the associated primes of $E_1 + E_2$ strictly contain the union of the associated primes of $E_1$ and $E_2$. There's a short exact sequence

$ 0 \to E_1 \to E_1 + E_2 \to (E_1 + E_2)/E_1 \to 0 $

where the last term is isomorphic to $E_2/(E_1 \cap E_2)$. So if they have trivial intersection the sequence splits and we have equality $\textrm{Ass}(E_1) \cup \textrm{Ass}(E_2) = \textrm{Ass}(E_1 + E_2)$. But I suspect that if the intersection is nontrivial there's a counterexample to this, I just can't think of what it would be. It won't work for something simple like $\mathbb{Z}$-modules, and I don't think it will work in general for integral domains. Or you might even need a non-Noetherian ring. I'm not sure.

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Let $R=k[x,y]/(x^2-xy, y^2-xy)$, $E_1=(x), E_2=(y)$

Then $Ass(E_1) = Ass (E_2) = (x-y)$ but $E_1+E_2$ contains the element $x-y$...