characterizations of integral
Let be a subring of a field , and let be a non-zero element of . The following conditions are equivalent:
is integral over .
belongs to .
is unit of .
Proof. Supposing the first condition that an equation
with ’s belonging to , holds. Dividing both by gives
One sees that belongs to the ring even being a unit of this (of course ). Therefore also the principal ideal of the ring coincides with this ring. Conversely, the last circumstance implies that is integral over .
- 1 Emil Artin: . Lecture notes. Mathematisches Institut, Göttingen (1959).
|Title||characterizations of integral|
|Date of creation||2013-03-22 14:56:54|
|Last modified on||2013-03-22 14:56:54|
|Last modified by||pahio (2872)|
|Synonym||characterisations of integral|