Integrally closed domain

An integral domain $𝑅$ with field of fractions $𝐾 = F r a c (𝑅)$ is integrally closed iff $𝛼 \in 𝐾$ is integral over $𝑅$ iff $𝛼 \in 𝑅$ . #m/def/ring This motivates the integral closure

―― 𝑅 = O F r a c (𝑅) : 𝑅

which in this case is the ring of integers of the field of fractions $𝐾$ . Thus $𝑅$ is integrally closed iff it equals its integral closure.

Integrally closed domain

See also