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 :

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?