Integrally closed

In mathematics, more specifically in abstract algebra, the concept of integrally closed has three meanings:

• A commutative ring $R$ contained in a commutative ring $S$ is said to be integrally closed in $S$ if $R$ is equal to the integral closure of $R$ in $S$.
• An integral domain $R$ 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.