Let $V$ be a finite-dimensional $F$-vector space and $T$ be an $F$-endomorphism on $V$. Let $\alpha$ be an $F$-basis for $V$. Then, there is a natural $F$-linear isomorphism $[-]_\alpha:End(V)\rightarrow M_n(F)$ where $dim(V)=n$.
Can this concept be extended to modules?
Let $R$ be a commutative ring and $M$ be a free $R$-module with the finite rank $n$. Let $\alpha$ be an $R$-basis for $M$.
Then, I guess that $[-]_\alpha:End_R(V)\rightarrow M_n(R)$ is an $R$-module isomorphism and we can define $\det(\phi)$ for $\phi\in End_R(V)$.
Is there a text introducing linear algebra concpets in the context of module? I'm reading Dummit&Foote, but I hate that he sticks with vector spaces for some concepts even though they can be greatly generalized.