Integral Extensions
One of the Cohen-Seidenberg theorems shows that there is a close relationship between the prime ideals of A and the prime ideals of B. Specifically, they show that an integral extension A⊆B has the going-up property, the lying over property, and the incomparability property. In particular, the Krull dimensions of A and B are the same.
When A, B are domains such that B is integral over A, A is a field if and only if B is a field. As a corollary, one has: given a prime ideal of B, is a maximal ideal of B if and only if is a maximal ideal of A. Another corollary: if is an algebraic extension, then any subring of L containing K is a field.
Let B be a ring that is integral over a subring A and k an algebraically closed field. If is a homomorphism, then f extends to a homomorphism .
Let be an integral extension of rings. Then the induced map
is closed. This is a geometric interpretation of the going-up property. This contrasts to the fact a flat morphism is open.
Let B be a ring and A its subring such that B is integral over A. If A is a Jacobson ring, then B is a Jacobson ring.
Read more about this topic: Integral Element
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