# Significance of integrally closed in an affinoid algebra

A Tate affinoid k-algebra is defined as a pair $$(R,R^+)$$ where $$R$$ is a Tate algebra and $$R^+$$ is an open and integrally closed subring of $$R$$ contained in the ring of powerbounded elements.

See for instance Defn 2.6 here.

What is the (geometric) significance of requiring $$R^+$$ to be integrally closed?

On a related note, what is the difference (geometrically) between total integral closure and integrally closed. It is surprisingly hard to find anything about total integral closures...

The adic spectrum of $$(R,R^+)$$ is a set of continuous valuations on $$R$$ having norm $$\leq 1$$ on $$R^+$$. It is clear from the definition and basic properties of integral closures that relaxing the integral closedness of $$R^+$$ buys you nothing new: a valuation on $$R$$ has norm $$\leq 1$$ on a subring $$R_0 \subset R$$ if and only if it has norm $$\leq 1$$ on the integral closure $$R^+$$ of $$R_0$$ in $$R$$. I believe this is the main reason one often assumes $$R^+$$ is integrally closed.
The difference between total integral closures and integral closures, roughly, corresponds to the difference between rank $$1$$ valuations and all valuations. But it might be better to ask a separate question about this as it is pure commutative algebra.