>>10761174Btw I'm asking because I have a local ring and a subring , and I want to show that the latter is local.
Let be the local ring of , and consider the prime ideal . I want to show that this is local - if was integral, this would be automatic.
I also have this further condition: is the subring of automorphisms from the action of some group . Now, if the group was finite, then would be integral over , but my group is not necessarily finite. Can we still conclude it?
I was thinking some argument like, let . Then is a unit in , so let . Acting with , we get on the one hand , and on the other, we get , hence , and so is a unit in .
This seems a little too easy. Is this right?
