Let $G$ be a finite group and $F$ be an algebraically closed field such that $char(F)$ does not divide $|G|$.
Then, Artin-Weddenburn theorem implies that there exist positive integers $n_1,...,n_k$ such that $F[G]\cong \prod_{i=1}^k Mat_{n_i}(F)$ as rings. How do I conclude that they are isomorphic as $F$-modules?
Texts I am reading say that "since they are isomorphic as rings, they are isomorphic as $F$-modules, hence $|G|=dim_F(F[G])=dim_F(\prod_{i=1}^k Mat_{n_i}(F))=\sum n_i^2$." However, to imply this, we must show that there exists an $F$-linear isomorphism which preserves given $F$-space structures. Since $F[G]$ and $\prod_{i=1}^n Mat_{n_i}(F)$ are isomorphic as rings, of course we can embed an $F$-space structure from one to another naturally induced by a ring isomorphism. However, dimension may differ and this does not mean that $F[G]$ and $\prod_{i=1}^k Mat_{n_i}(F)$ are isomorphic as $F$-spaces where the $F$-space structures are already given.
How do I prove this? Thank you in advance.