Modules isomorphism

Question

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!

Answer 1

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).

Answer 2

The simplest non-field rings are principal ideal domains, for which there is a structure theorem, at least for finitely generated modules.

For example, a finitely generated $\mathbb{Z}$-module (an abelian group) is isomorphic to a direct sum of copies of $\mathbb{Z}$ and $\mathbb{Z}/n\mathbb{Z}$ for various $n$.

Despite PIDs being a very restricted class of rings, this theorem still has fairly broad application. For example, applying it to the PID $k[X]$ gives the rational and Jordan forms for matrices.

For modules that are not finitely generated, the situation can be quite complicated, even for PIDs. Take the abelian groups $\mathbb{Q}$ and $\mathbb{Z}_p = \varprojlim \mathbb{Z}/p^k$. They are both torsion-free but not free, and not so easy to capture in a general structure theorem. In fact, there are still unanswered questions regarding abelian groups of rank $>1$.

For non-PIDs, the situation may be difficult even for finitely-generated modules. The failure of unique factorization in Dedekind rings, for example, corresponds to a failure to uniquely decompose modules into irreducibles. This and other failures point towards the study of abstract properties of modules, e.g. projectivity, flatness.

There are other structure theorems, but it would take a very large amount of space to cover a small percentage of them. For example, if $G$ is a group, then a $k[G]$-module is just a $k$-representation of $G$. If $G$ is finite, and $k$ has characteristic $0$, then there is a complete theory of finite-dimensional representations of $G$. But if $k$ has positive characteristic, then we are in the domain of modular representation theory, which is much more difficult and even an active, current area of research!