Originally Posted by

**Ant** Hi,

I believe the answer is yes. As you say, R is an ideal of R hence if R is noetherian then R must be finitely generated.

In general, when you consider R as an ideal of R then it is certainly finitely generated (simply take 1 as a generating set!).

So yes, fields only have two ideals both of which are finitely generated; {0} is an obvious generating set for the ideal {0}, and {1} is a generating set for R.

Of course R need not be finite in size!

It's been a little while since I've covered this material so don't take the above as gospel. Good luck!