Modules isomorphism Studying vector spaces, we can find the well known result that every vector space of dimension $n$ over a field $k$ is isomorphic to $k^n$. Is there a similar theorem for modules?
Thanks guys!

Not really, as a module doesn't necessarily have a basis.  What can be said is that for a module M which has a system of $n$ generators, there exists a surjective homomorphism from $A^n$ to $M$, that sends each vector of the canonical basis of $A^ n$ to one of the generators of $M$.
Actually, if A is a ring, a module which is isomorphic to some $A^n$ is called a finitely generated free A-module. This is not the case in general. For instance, a finite abelian group $G$ is a $\mathbf Z$-module which is certainly not free, since there exists an integer $n$ such that (in additive notation) $ng=0$ for every $g\in G$.
Worse, there exist  non-commutative rings such that $A^m$ is isomorphic to $A^n$ for some $m \neq n$. This is not true for commutative rings and for non-commutative fields (such as the quaternions).
