Commutative Algebra/Integral dependence

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Definition (integral dependence):

Let be a commutative ring, and let be a subring, so that is a ring extension. An element is called integral over if there exists a monic polynomial with coefficients in ,


such that .

Proposition (criteria for ):


Proposition (polynomials of integral elements are again integral):

Let be a ring extension, integral over , . Then is integral over .

Proof: Let be a ring extension of such that decomposes into linear factors in ; such an extension always exists.