View source for Commutative Algebra/Integral dependence

{{definition|integral dependence|Let <math>S</math> be a commutative ring, and let <math>R \subset S</math> be a subring, so that <math>S/R</math> is a ring extension. An element <math>s \in S</math> is called '''integral''' over <math>R</math> if there exists a monic polynomial with coefficients in <math>R</math>,
:<math>p(x) = x^n + a_{n-1} x^{n-1} + \cdots + a_1 x + a_0</math>, <math>\forall j \in \{0, \ldots, n\}: a_j \in R</math>
such that <math>p(s) = 0</math>.}}

{{proposition|criteria for }}

{{proposition|polynomials of integral elements are again integral|Let <math>S/R</math> be a ring extension, <math>s_1, \ldots, s_k \in S</math> integral over <math>R</math>, <math>p(x_1, \ldots, x_n) \in R[x_1, \ldots, x_n]</math>. Then <math>p(s_1, \ldots, s_k)</math> is integral over <math>R</math>.}}

{{proof|Let <math>\overline S</math> be a ring extension of <math>S</math> such that <math>p</math> decomposes into linear factors in <math>\overline S</math>; such an extension always exists.}}

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