Integral closure of a subring

Definition
Let $$R$$ be a unital subring of a commutative unital ring $$S$$. The integral closure of R in $$S$$ is defined as the set of those elements of $$S$$ that are integral over $$R$$, viz that satisfy monic polynomials over $$R$$.

If $$R$$ equals its integral closure, we call it an integrally closed subring and if the itnegral closure of $$R$$ equals $$S$$, we call it an integrally dense subring.