prime ideals by Artin are prime ideals
Theorem. Due to Artin, a prime ideal of a commutative ring is the maximal element among the ideals not intersecting a multiplicative subset of . This is equivalent (http://planetmath.org/Equivalent3) to the usual criterion
of prime ideal (see the entry prime ideal (http://planetmath.org/PrimeIdeal)).
Proof. . Let be a prime ideal by Artin, corresponding the semigroup , and let the ring product belong to . Assume, contrary to the assertion, that neither of and lies in . When generally means the least ideal containing and an element , the antithesis implies that
whence by the maximality of we have
Therefore we can chose such elements of (N.B. the multiples) that
This is however impossible, since the product belongs to the semigroup and . Because the antithesis thus is wrong, we must have or .
. Let us then suppose that an ideal satisfies the condition (1) for all
. It means that the set is a multiplicative semigroup. Accordingly, the is the greatest ideal not intersecting the semigroup , Q.E.D.
Remark. It follows easily from the theorem, that if is a prime ideal of the commutative ring and is a subring of , then is a prime ideal of .
|Title||prime ideals by Artin are prime ideals|
|Date of creation||2013-03-22 18:44:55|
|Last modified on||2013-03-22 18:44:55|
|Last modified by||pahio (2872)|