Hi everyone. I'm preparing for a test tomorrow and there are concepts and methods, which I seem to have forgotten or never understood properly.

For eksample there are these exercises, which I'm having difficulties with, if someone could help me throught them, it would be nice (Nod) :

1) Explain why the -modules and have finite lenght and write a chain of composition.

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

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

4) Are some of the three -modules

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

Now, in 1) I would say that and therefore

correct?

and so so ? So thus the lenght is three in both cases and thus finite?

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?