integral closure is ring
Proof. Let be an arbitrary element of the integral closure of in . Then there are the elements of such that
where . If is a polynomial in with degree , we have
where the elements belong to . This procedure may be repeated until we see that is an element of the -module generated by . Accordingly,
is a finitely generated -module.
Now we have evidently . Let be another element of . Then
is a finitely generated -module, whence is a finitely generated -module. Because the elements and belong to , they are integral over and thus belong to . Consequently, is a subring of (see the http://planetmath.org/node/2738subring condition).
- 1 M. Larsen & P. McCarthy: Multiplicative theory of ideals. Academic Press, New York (1971).
|Title||integral closure is ring|
|Date of creation||2013-03-22 19:15:40|
|Last modified on||2013-03-22 19:15:40|
|Last modified by||pahio (2872)|