I have a nice solution to the following problem and I thought of writing a paper about it but beforehand, I wanted to ask the problem here to see if this is an easy problem and if you people can solve it easily. If its easy then I will not write a paper about my solution. I know of many people who did not manage to solve the problem so lets see if you can.
Let A be a commutative noetherian algebra. Let M be a finitely generated A-module. For every finitely generated A-module N we define SuppN = V(Ann(N)) \subset Spec(A).
Spec(A) denotes the prime ideals of A.
Ann(N) denotes all the members a of A such that aN=0.
Prove that for any prime ideal p in SuppM, M has a quotient isomorphic to A/p. Meaning there is an A-module N, which is a submodule of M such that M/N is isomorphic to A/p.
Good Luck :)