Finding simple modules in fields

Question

I am struggling with finding all simple are modules for general rings

i know that the simple modules of R correspond with the simple modules of R/rad(R), where rad(R) is the Jacobson radical but i am struggling to apply this to say polynomial rings over complex C[X]

or defining a ring of real valued CTS functions

any help on this would be greatly appreciated

Answer 1

This is a very difficult problem that has no known general solution. The following link, execise 17G asks the reader to prove that there are infinitely many pairwise non-isomorphic unfaithful simple modules over a polynomial ring in one variable with coefficients in a division ring. If you are interested in studying the simple modules over a particular family of rings I recommend looking up literature on the specific rings if you are having trouble classifying the simple modules on your own.