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

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

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