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
Famous quotes containing the words integral and/or extensions:
“Painting myself for others, I have painted my inward self with colors clearer than my original ones. I have no more made my book than my book has made me—a book consubstantial with its author, concerned with my own self, an integral part of my life; not concerned with some third-hand, extraneous purpose, like all other books.”
—Michel de Montaigne (1533–1592)
“The psychological umbilical cord is more difficult to cut than the real one. We experience our children as extensions of ourselves, and we feel as though their behavior is an expression of something within us...instead of an expression of something in them. We see in our children our own reflection, and when we don’t like what we see, we feel angry at the reflection.”
—Elaine Heffner (20th century)