# 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 )**:

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**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.