Let $K$ be a field and let $A$ be a $K$-algebra which is finite dimensional as $K$-vector space. Then the nice structure theorem for artinian rings says that we can write $A$ as the direct product of local $K$-algebras. Is there also a structure theorem for finitely generated modules over such $A$?

If this is too vague, my naive hope would be that, similar to the case of principal ideal domains, we can write every finitely generated $A$-module $M$ as $$\oplus_{k=1}^n I_k/J_k$$ where $J_k\subseteq I_k$ are ideals of $A$ and $I_k/J_k$ should denote the ideal in $A/J_k$ given by all $x+J_k$ for $x\in I_k$.