definition of prime ideal by Artin
Lemma. Let be a commutative ring and a multiplicative semigroup consisting of a subset of . If there exist http://planetmath.org/node/371ideals of which are disjoint with , then the set of all such ideals has a maximal element with respect to the set inclusion.
Proof. Let be an arbitrary chain in . Then the union
which belongs to , may be taken for the upper bound of , since it clearly is an ideal of and disjoint with . Because thus is inductively ordered with respect to “”, our assertion follows from Zorn’s lemma.
Definition. The maximal elements in the Lemma are prime ideals of the commutative ring.
- 1 Emil Artin: Theory of Algebraic Numbers. Lecture notes. Mathematisches Institut, Göttingen (1959).
|Title||definition of prime ideal by Artin|
|Date of creation||2013-03-22 18:44:31|
|Last modified on||2013-03-22 18:44:31|
|Last modified by||pahio (2872)|