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.