I'm currently going through a proof and I've come across something I don't really understand:
Next, an endomorphism of a left $A$-module $M$, over a ring $A$ is an $A$- homomorphism $M\rightarrow M$; these form a ring End$_A(M)$, where addition is given by the rule $(\phi + \psi)(x):=\phi (x)+\psi (x)$ and multiplication by composition of maps. If $A$ is a $k$-algebra, then so is End$_A(M)$, for multiplication by an element of $k$ defines an element in the centre of End$_A(M)$. In the case when $A$ is a division algebra, $M$ is a left vector space over $A$, so the usual argument from linear algebra shows that choosing a basis of $M$ induces an isomorphism End$_A(M)\cong M_n(A)$, where $n$ is the dimension of $M$ over $A$.
I don't understand the "usual argument from linear algebra" part. I don't know what argument it might be referring to. I understood well up until that point. I'm assuming that $M_n(A)$ are the $n\times n$ matrices with coefficients from $A$. To be honest I'm not sure what the difference between a left vector space and a left module is. I thought that vector spaces were only defined over fields, where the concepts of left and right coincide.
If anyone could clarify, I'd be extremely grateful, as proof that follows this is essential to my dissertation. Thanks for any replies!