Prime Ideals (Pt. IV)
Point of Post: This is a continuation of this post.
We now record some of the basic miscellaneous facts concerning prime ideals, things like what the spectrum of a product looks like and that morphisms pull back prime ideals:
Theorem: Let be a collection of commutative unital rings. Then,
Proof: This follows immediately from the characterization of prime ideals as being those whose quotient ring is an integral domain, the fact that the ideals of a product of commutative unital rings are of the form , and the observation that .
Theorem: Let and be commutative rings and a morphism. Then, if one has that or .
Proof: Suppose that then and so and thus . The conclusion readily follows.
By considering the inclusion map from a subring into a ring we get the following corollary:
Corollary: Let be a commutative ring, a subring, and . Then, .
We last make an obvious note about how a prime ideal can end up in a product of ideals which mirrors precisely the case for the primes in :
Theorem: Let be a commutative ring, ideals of . If and then for some .
Proof: By way of contradiction assume that there exists some fore ach , then by assumption that is prime we find that which is an evident contradiction.
1. Dummit, David Steven., and Richard M. Foote. Abstract Algebra. Hoboken, NJ: Wiley, 2004. Print.
2. Bhattacharya, P. B., S. K. Jain, and S. R. Nagpaul. Basic Abstract Algebra. Cambridge [Cambridgeshire: Cambridge UP, 1986. Print.
3. Reid, Miles. Undergraduate Commutative Algebra. Cambridge: Cambridge UP, 1995. Print.