Suppose that $k$ is an algebraically closed field. Let $R$ be a noncommutative $k$-algebra with $\dim_k R=7$. What are the vector space dimensions of the simple components of $R$? Use this to find the isomorphism classes of simple $R$-modules.
It is obvious that the dimensions of the simple components must be finite. Since maps between simple modules are zero maps or isomorphisms by Schur's Lemma, once one knows the vector space dimension of the simple modules, you can give a description of the simple modules themselves and the isomorphism classes easily based on their dimension. The part I am confused about is how to find these dimensions to begin with. I know this probably makes use of the Artin-Wedderburn Theorem. Since $R$ is semisimple, it is (isomorphic to) a product of matrix rings over division rings. But I do not see exactly what this gets me in relation to the fact that $\dim_k R=7$. How do I proceed with this problem?