Let $R$ be a commutative $k$-algebra, which is an integral domain, where $k$ is a field of characteristic zero. Let $R_i$, $i \in \mathbb{N}$, be $k$-subalgebras of $R$. Denote $A:=\cap_{i \in \mathbb{N}}R_i$. Trivially, $A$ is a $k$-subalgebra of each $R_i$ and of $R$.
I am interested in a list of 'good' properties $P$ such that if each of $R_i$ satisfies property $P$, then $A$ also satisfies property $P$.
Examples for such good properties $P$: (1) Normality (integral closedness in its field of fractions). (2) UFD.
Additional properties that I am interested in, but I do not know if they are 'good': beign Noetherian, being Artinian.
What if, in addition, $R_i$ is a descending chain, namely, if $i < j$, then $R_i \supseteq R_j$.