Let $R$ be a commutative domain and let $\theta$ be a ring automorphism of $R$. The fixed ring of $\theta$ is defined by $R^{\theta}:=\{r \in R: \ \theta(r)=r \}.$ An ideal $I$ of $R$ is called invariant if $\theta(I)=I$. We say that $R$ has a large fixed ring if $I \cap R^{\theta} \neq (0)$ for all non-zero invariant ideals of $R$.
Question: Is there any (partial) characterization of commutative noetherian domains that have large fixed rings for all their infinite order automorphisms?
I should add that if $R$ is a commutative domain and $\theta$ is an automorphism of finite order of $R$ such that the order of $\theta$ is a non-zero element of $R,$ then $R$ has a large fixed ring (a trivial consequence of the Bergman-Isaac theorem).
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$\begingroup$ This class contains all the rings which have no infinite order automorphisms (for example the ring of integers of any number field, the ring of integers of any p-adic field, any p-adic field, any number field, the integers in the field obtained by adjoining the square root of every rational integer to the rationals, the real numbers...). I'm not sure you're going to get a classification! $\endgroup$– ericCommented Dec 16, 2015 at 19:16
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