Let $A$ be an associative (unital) algebra. Let $M_1,\cdots, M_r$ be pairwise non-isomorphic simple $A$-modules and let $V=\bigoplus^r_{i=1}V_i$, where $$ V_i=M_{i,1}\oplus \cdots\oplus M_{i,n_i}\quad \text{ with } \quad M_{i,1}\cong \cdots\cong M_{i,n_i}\cong M_i $$for some $n_i\in \mathbb{N}$.
Let $X\subseteq V$ be an $A$-submodule.
Question: Is it always true that $X=\bigoplus^r_{i=1}X_i$ with each $X_i$ an $A$-submodule of $V_i$ ?
(In fact, I only need the case where $A$ is an algebra over a field $K$ and $\dim_KV<\infty$.)
