*PM : separability is required for integral closures to be finitely generated, id = 9324 new !-- WP guess : separability is required for integral closures to be finitely generated-- Status:
22.
*PM : separability is required for integral closures to be finitely generated, id = 9324 new !-- WP guess : separability is required for integral closures to be finitely generated-- Status:
23.
Later, C . R . Leedham-Green showed that such an " R " may constructed as the integral closure of a PID in a quadratic field extension ( Leedham-Green 1972 ).
24.
One important consequence of the theorem is that the integral closure of a Dedekind domain " A " in a finite extension of the field of fractions of " A " is again a Dedekind domain.
25.
In general, the integral closure of a Dedekind domain in an infinite algebraic extension is a Pr�fer domain; it turns out that the ring of algebraic integers is slightly more special than this : it is a B�zout domain.
26.
Samuel were sufficiently taken by this construction to pose as a question whether every Dedekind domain arises in such a fashion, i . e ., by starting with a PID and taking the integral closure in a finite degree field extension.
27.
In the latter example the ring can be made into an UFD by taking its integral closure in ( the ring of Dirichlet integers ), over which becomes reducible, but in the former example " R " is already integrally closed.
28.
If the situation is as above but the extension " L " of " K " is algebraic of infinite degree, then it is still possible for the integral closure " S " of " R " in " L " to be a Dedekind domain, but it is not guaranteed.
29.
1 ?! 2 results immediately from the preservation of integral closure under localization; 2 ?! 3 is trivial; 3 ?! 1 results from the preservation of integral closure under localization, the exactness of localization, and the property that an " A "-module " M " is zero if and only if its localization with respect to every maximal ideal is zero.
30.
1 ?! 2 results immediately from the preservation of integral closure under localization; 2 ?! 3 is trivial; 3 ?! 1 results from the preservation of integral closure under localization, the exactness of localization, and the property that an " A "-module " M " is zero if and only if its localization with respect to every maximal ideal is zero.
