Let $A\subset R$ be commutative rings with identity. $x\in R$ is integral over $A$ if $x$ is vanished by some monic polynomial with coefficients in $A$. Define the integral closure $\mathrm{Cl}_R(A)$ of $A$ in $R$ to be the set of integral elements of $R$ in $A$. If we are only talking about the subrings of $R$ we abbreviate the notation as $\overline{A}$. $A$ is integrally dense in $R$ if $\overline{A}=R$. $A$ is integrally closed in $R$ if $\overline{A}=A$.
Note that we can generalize the defintion of integral closure to any subsets of $R$.
Here are some standard results using this notation: (Fix the ring $R$ so we suppose all other sets are subsets of $R$.) We have
$(i)$ $A\subset \overline{A}$ for any set $A$.
$(ii)$ $\overline{\emptyset}=\emptyset$.
$(iii)$ $\overline{(\overline{A})}=\overline{A}$ for any ring $A$.
$(iv)$ $\overline{A\cup B}=\overline{A}\cup\overline{B}$ for any rings $A, B$.
So the integral closure of a special class of $P(R)$(namely the family of the subrings of $R$, but note that this family is not closed under unions, but we can focus on a chain of subrings instead) satisfies the Kuratowski closure axioms. I would like to know if this has been studied before.
For results relating to integrally dense we have:
Let $A\subset B \subset R$ be rings. Then
$(i)$ If $x\in \mathrm{Cl}_R(B)$ and $A$ is integrally dense in $B$ then $x\in \mathrm{Cl}_R(A)$.
This implies that
$(ii)$ If $A$ is integrally dense in $B$ and $B$ is integrally dense in $R$, then $A$ is integrally dense in $R$.
Does all the above definitions and results have any significance in other areas?(like algebraic geometry or algebraic number theory, etc)
In the special case of ideals $\mathfrak{a}\subset R$. Radical ideals are closed.