I'm stuck on something, I'd appreciate it if you could help :)
Let $A$ be a $F$-algebra, $V$ be a completely reducible $A$-module and $M$ is an irreducible $A$- module. Now let's consider $M(V)$, which is the homogenous part of $V$. ( Recall: $M(V)$: sum of submodules of $V$ isomorphic to the irreducible $M$.) My question is the following :
How can we prove that "If $U$ is a submodule of $V$, then $M(U) = U \cap M(V)$"
My attempt : Actually, the left side is obvious because any submodule of $U$ which isomorphic to $M$ is also a submodule of $V$ isomorphic to M, and so $ M(U) \subset M(V)$. How can we get the other side ?