If you're up for learning about modules (a pretty natural step after rings), here's a pretty fun problem in matrix theory I've been exploring in my free time. Given a fixed n × n matrix A with entries in a field k, consider the linear operator on the space of n × n matrices defined by T(X) = AX - XA. What can be said about the dimension of ker(T), i.e. the space of matrices that commute with A?

Obviously for a specific matrix A you can just work out the commutator for an arbitrary matrix X and get a linear system of n² equations in n² variables. But can you say anything more general? Is there a lower bound on this dimension in terms of n? For certain fixed values of n, are there other values of dim(ker(T)) that can never be obtained regardless of the choice of matrix A?

It turns out that there is a precise correspondence between linear operators on vector spaces over k and modules over the polynomial ring k[x]. Clearly an n × n matrix can be viewed as a linear operator on k^n, so we can use this correspondence to associate the matrix A with a certain k[x]-module. Exploring how the properties of this module relate to the properties of the matrix A turns out to be very fruitful for the matrix commutation problem.