# Integrally closed subring

From Commalg

This article defines a property that can be evaluated for a unital subring in a commutative unital ring: given any commutative unital ring and a subring thereof, the property is either true or false for the pair

View a complete list of such properties

## Definition

### Symbol-free definition

A unital subring of a commutative unital ring is said to be **integrally closed** in the ring if any element of the ring integral over the subring (i.e. satisfying a monic polynomial over the subring) must lie inside the subring itself.

## Related notions

### Related ring properties

- Normal ring is a ring that is integrally closed in its total quotient ring
- Normal domain is an integral domain that is integrally closed in its field of fractions