Given a ring $A$ with identity we can define a ring of all its subrings, with addition and multiplication defined by $S+R=\left<\{s+r: s\in S, r\in R\}\right>$ and $SR=\left<\{sr:s\in S, r\in R\}\right>$ In this way the multiplication and addition behave very badly.
So is there another nice way to keep the addition information from the original ring $A$ but also giving nice properties for addition of subrings?(The most important one is the additive inverse).
As practice let’s say we have defined a nice structure for it, say $K$ is the ring of all its subrings with all suitable operations. Now let’s view integral closure as a “homomorphism” from $K$ to itself. A subring is integrally closed if it maps to itself and a $A$ is integral over a subring $R$ if it maps to $A$(if $A$ by any chance is the addition identity after a suitable definition for addition we can describe the set of all subrings which $A$ is integral over them as the kernel of the integral closure homomorphism.)
Add: As Alex pointed out, the multiplication identity can be canonically given by the image of the homomorphism $h: \mathbb{Z}\to A$ defined in the obvious way.