In mathematics, more specifically in abstract algebra, the concept of integrally closed has three meanings: A commutative ring contained in a commutative ring is said to be integrally closed in if is equal to the integral closure of in . An integral domain 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.