Integrally closed

In mathematics, more specifically in abstract algebra, the concept of integrally closed has two meanings, one for groups and one for rings.

A commutative ring ${\displaystyle R}$ contained in a 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 ${\displaystyle S}$. That is, for every monic polynomial f with coefficients in ${\displaystyle R}$, every root of f belonging to S also belongs to ${\displaystyle R}$. Typically if one refers to a domain being integrally closed without reference to an overring, it is meant that the ring is integrally closed in its field of fractions.

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