## Prime Ideals (Pt. IV)

**Point of Post: **This is a continuation of this post.

*Miscellaneous Facts*

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, .*

\text{ }$

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.

**References:**

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.

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