# Commutative Algebra/Integral dependence

Definition (integral dependence):

Let ${\displaystyle S}$ be a commutative ring, and let ${\displaystyle R\subset S}$ be a subring, so that ${\displaystyle S/R}$ is a ring extension. An element ${\displaystyle s\in S}$ is called integral over ${\displaystyle R}$ if there exists a monic polynomial with coefficients in ${\displaystyle R}$,

${\displaystyle p(x)=x^{n}+a_{n-1}x^{n-1}+\cdots +a_{1}x+a_{0}}$, ${\displaystyle \forall j\in \{0,\ldots ,n\}:a_{j}\in R}$

such that ${\displaystyle p(s)=0}$.

Proposition (criteria for ):

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Proposition (polynomials of integral elements are again integral):

Let ${\displaystyle S/R}$ be a ring extension, ${\displaystyle s_{1},\ldots ,s_{k}\in S}$ integral over ${\displaystyle R}$, ${\displaystyle p(x_{1},\ldots ,x_{n})\in R[x_{1},\ldots ,x_{n}]}$. Then ${\displaystyle p(s_{1},\ldots ,s_{k})}$ is integral over ${\displaystyle R}$.

Proof: Let ${\displaystyle {\overline {S}}}$ be a ring extension of ${\displaystyle S}$ such that ${\displaystyle p}$ decomposes into linear factors in ${\displaystyle {\overline {S}}}$; such an extension always exists. ${\displaystyle \Box }$