well, they're both finite modules and thus obviously Artinian (and Noetherian). so they have finite length. your solution for is wrong.

recall that so and

2) Find the associated primes and the support of the above -modules

let be the set of associated primes of R module M. if is a prime number, then so

also

is a PID. so a prime ideal of R is in the form where is an irreducible element of and but if and only if which is possible

3) k is a field. Explain why is the only prime ideal in . Is a simple R-module?

only if because is irreducible. the answer to the second part of your question is no because is a non-zero submodule of

the only possible isomorphism here is that "as rings" we have the isomorphism sends to and to to see that there are no other isomorphisms let

4) Are some of the three -modules

, , isomorphic? Are some of them isomorphic as rings?

In (4) I would say that the two last are isomorphic, because you can make an ismomorphism from to by sending X to Y, Y to X and 1 to 1. Correct?

suppose is a "ring" isomorphism. let then

so and thus (why?) therefore and, since is an isomorphism, we get that means which is clearly false. so there's no

such isomorphism. let me show you how to prove that and are not isomorphic as "R-mouldes": suppose is an R-module isomorphism. let

then so and hence therefore which gives us that means

which is obviously nonsense!