@DanielFischer hi

ho

can you take a look on some definition i saw in a book? something dont work there for me

What is it about?

rings

Okay, I may understand the definition then.

im not sure what's $gx$

i mean, $g$ acts on $R$ but $gx$ is an automorphism doesn't it?

There's a typo in the definition, it should be $gx = x$ for all $g \in G$. It's a common abuse of notation to denote the automorphism that is the image of $g$ by $g$ too. So "$g$" is the automorphism, and $gx$ is the image of $x$ under the automorphism.

If we use full unabused but cumbersome notation, $\varphi \colon G \to \operatorname{Aut} R$ is the homomorphism by which $G$ acts on $R$.

And $R^G = \{ x \in R : (\forall g \in G)(\varphi(g)(x) = x)\}$.

@DanielFischer now that make sense!

thanks ^^

you're welcome

so that's the stabilizer

No, the stabiliser (of some element or subset of $R$) would be a subgroup of $G$. $R^G$ is the "fixed ring" of $G$, like the fixed field of a subgroup of the Galois group of a Galois extension.

Ah. ok

commutative algebra is going to be like what we learned in Galios theory ?

It's probably going to use some Galois theory. But it's a different beast overall.

hm.

would you say it's more interesting?

I don't know. Never really learnt the stuff. Algebra is not so much my cup of tea.

i would say its not mine either , but given the courses i showed you, there is not way around it ^^