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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.

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    To show only if: Show that a direct sum of two isomorphic simple modules does not have the property ((Morita) equivalently show that a vector space of dimension 2 does not have the property).2012-01-16

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