In mathematics, more specifically in abstract algebra, the concept of integrally closed has three meanings:
- A commutative ring <math>R</math> contained in a commutative ring <math>S</math> is said to be integrally closed in <math>S</math> if <math>R</math> is equal to the integral closure of <math>R</math> in <math>S</math>.
- An integral domain <math>R</math> is said to be integrally closed if it is equal to its integral closure in its field of fractions.
- An ordered group G is called integrally closed if for all elements a and b of G, if an ≤ b for all natural numbers n then a ≤ 1.