Exercise on distributive module lattices I'm trying to do the very first exercise in Representations and cohomology I by Dave Benson, it's been bugging me for a while now. I don't really know how to start, although I imagine we will need to induct on the composition length of $M$ and use the Zassenhaus isomorphism theorem.
Exercise Let $A$ be a unital ring. Suppose that $M$ is a module of finite composition length. Show that the submodules of $M$ satisfy the distributive laws
$$(A+B)\cap C=(A\cap C)+(B\cap C)$$and$$(A\cap B)+C=(A+C)\cap (B+C)$$if and only if $M$ has no subquotient isomorphic to a direct sum of two isomorphic simple modules. 
 A: EDIT: I hope this makes more clear my answer.
It is known that a modular lattice is distributive if, and only if, its Hasse  diagram does not include a diamond lattice. See for example the beginning of Chapter II, Distributive Lattices, of General Lattice Theory, by George A. Gratzer.
Let $\mathcal{S}(M)$ be the modular lattice of submodules of a module $M$. The Correspondence Theorem for submodules says that the lattice $\mathcal{S}(M/N)$ of submodules of a quotient $M/N$ is isomorphic to the lattice {$L\leq M\mid N\leq L$}, i.e. the part of the lattice $\mathcal{S}(M)$ which is above the element $N$.
Notice that if $M$ has composition series, then every submodule and every quotient also has composition series.
Suppose that $\mathcal{S}(M)$ is not distributive. Then we can find a diamond diagram somewhere in this lattice, with a submodule $N$ on the bottom, $L$ on the top and $S,T$ and $U$ in the middle. Therefore in the lattice $\mathcal{S}(M/N)$, if $L'=L/N, S'=S/N,T'=T/N$ and $U'=U/N$ then $L'=S'\oplus T'=S'\oplus U'=T'\oplus U'$. Now $L'$ also has a composition series, so by the Jordan-Hölder Theorem we must have that two of the three simple modules $S', T', U'$ are isomorphic. Therefore $M$ has a subquotient, namely $L/N$, which is isomorphic to, say, $S'\oplus S'$.
Conversely, if there exists such subquotient $L/N$ then it is the direct sum of two simple submodules, both isomorphic to a simple module $S$. In fact there is a third submodule, which is isomorphic to {$(x,x)\in S\oplus S\mid x\in S$}. Therefore we have a diamond diagram in $\mathcal{S}(M/N)$, which, again by the Correspondence Theorem also appears in $\mathcal{S}(M)$, between $N$ and $L$. This means that the lattice is not distributive.
