Ring theory, graded rings and noetherian rings
Hi, I need help with this question. Let be a graded ring (a ring with a sequence of abelian subgroups of such that for each , , and = the direct sum of the abelian groups). I need to show that if is right noetherian then is a finitely generated right ideal of .
So far I have gathered the following:
A ring is right noetherian if it is noetherian as a right R-module.
If a module is noetherian then it and all its submodules are finitely generated.
So I thought if I could show that as a right -module is a submodule of as a right R-module then it would show it is finitely generated, but I'm not sure how to do this (or even if it's true). And then I'm not too sure about showing the right ideal part.
Could anyone point me in the right direction?