Integral closure

The integral closure of $R$ in an $R$-algebra $A$ is the subring of elements of $A$ integral over $R$.

If a graded ring $A$ is a graded extension of a graded subring $R$, then the integral closure of $R$ in $A$ is graded: every homogeneous component of an integral element is integral. This follows by separating the extremal degrees in a monic integral equation and inducting on the number of nonzero homogeneous components.