Talk:Fixed-point subring

"More general" definition

For the definition given in the article,

"More generally, if G is a group acting on R, then the subring of R

$R^{G}=\{r\in R\mid g\cdot r=r,\,g\in G\}$

is called the fixed subring or the ring of invariants",

the group action property (i.e. the properties (gh).r = g.(h.r) and 1.r = r) is not needed for R^G to be a subring – indeed, the fact that G is a group isn't even used (the multiplication in G is never used, at least not in proving that R^G is a ring); all that is used is that for each g in G, the map r → g.r is a ring homomorphism. Therefore the more appropriate context would seem to be that the group G acting on R should be replaced by just a set S of ring automorphisms/endomorphisms of R, so I have added a {citation needed} tag requesting a reference to the definition given in the article.

My guess would be that although the group structure and group action structure are not needed in the definition of R^G, they are useful beyond this – e.g., in the situations that rings of invariants arise we probably almost always have such a group action. However I am not familiar with invariant theory so I'm mentioning this just in case the definition in the article is faulty or is given at the incorrect level of generality. Joel Brennan (talk) 20:24, 30 May 2022 (UTC)

As one considers only the action of automorphisms, a set S acting on R is a subset of the automorphism group of R. So, the fixed points of S are exactly the fixed points of the group generated by S. It follows that restricting the theory to group actions is not really a restriction. D.Lazard (talk) 10:40, 31 May 2022 (UTC)

(D.Lazard has already answered the question). Somehow tangentially related matter: shouldn't the article called "ring of invariants" instead of fixed-point subring. While accurate and less ambiguous, "fixed-point subring" is less common a term. -- Taku (talk) 06:35, 1 June 2022 (UTC)

The article consider two related concepts: the "fixed subring" of a ring automorphism, and the "ring of invariants" of a group of ring automorphisms. The use of "point" in the title is (at least) astonishing, as there is no point in the article, unless one call point every element of every ring. So, I agree with the sugggested move, and suggest also to replace "fixed-point subring" with "fixed subring" everywhere in the article. D.Lazard (talk) 08:43, 1 June 2022 (UTC)