Integral closure of a subring

Definition

Let $R$ be a unital subring of a commutative unital ring $S$. The integral closure of <amth>R</math> 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.