Commutative Algebra/Integral dependence

Definition (integral dependence):

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

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

such that $p(s)=0$.

Proposition (criteria for ):

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Proposition (polynomials of integral elements are again integral):

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

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