I am trying to use Zorn's lemma to prove that the set of all prime ideals in a commutative ring with unity has a minimal element. I have already seen the other posts on the site asking for help proving the same theorem - so please finish reading before marking my question as a duplicate. My question here is more about how to use Zorn's lemma. I have a specific step that seems to be tripping me up.
So heres what I got: 1. Define the set of all prime ideals of some non-zero commutative ring with unity, then order this set with respect to reverse inclusion.
Since I have at my disposal theorems that 1) every such ring has a maximal ideal, and 2) every maximal ideal is a prime ideal, I know that my set of prime ideals is non empty. Further I have proven that reverse inclusion is a partial order on my non empty set of prime ideals.
We have gathered all the ingredients we need to apply Zorn's lemma - except the chain condition, and this is where I have a problem. If I let $T$ be an arbitrary chain, I can use intersections of prime ideals to produce my maximal element (in my case the minimal). But I have a problem, What if $T = \emptyset$. It appears since I am letting $T$ be an arbitrary chain, and $\emptyset$ appears to vacuously be an acceptable chain, so whatever my proof does should hold. But then I need some prime ideal $P$ such that $P \subseteq \emptyset$? It seems like this would force $P$ to also be the empty set which wont work for me since the empty set cant be an ideal.
I've been known to big deal minor issues - thats probably what I am doing here, but I cant move on until I have some closure here. Thanks for any help.