TITLE: Factorial Rings and The Axiom of Choice
QUESTION [9 upvotes]: It is shown in Lang's Algebra (and many other books I assume) that:
if A if a principal entire ring, then A is a factorial ring.
The proof uses Zorn's Lemma. Is this theorem equivalent to the axiom of choice?

REPLY [6 votes]: This is an extended comment on Keith's answer:
Another way to think about Keith's argument (or the more general argument for the existence of minimal primes in Noetherian rings) is that one constructs a tree as follows: beginning 
with an ideal $I$, if it not prime then we may find $J_1, J_2$ properly containing $I$
such that $J_1 J_2 \subset I$, and we make a "factorization tree" joining $I$ to $J_1$
and $J_2$, and then iterate this procedure.
Noetherianness says that each branch of the tree has finite length, and since each
edge has finite valence, the whole tree is finite, hence we eventually must reach prime
ideals, and we find a finite collection of prime ideals $P_i$ whose product is contained
in $I$.
The fact about trees being used can be phrased (in the contrapositive) as:
an infinite tree with every vertex of finite valence has an infinite branch.  (This has a name, which I forget; it is also what underlies the proof of Bolzano--Weierstrass.)
Now the proof of this fact uses DC (as far as I can tell), and it's probably well-known
to logicians exactly whether or not it is equivalent to DC (or, at any rate,
exactly how strong it is).
The question then becomes whether we can model any tree (whose internal vertices have
valence 3) as a factorization tree (in a PID, say, although the possibility of allowing
more general Noetherian rings seems reasonable, at least if it helps to give a positive
answer).
Added: The result on trees is called Koenig's lemma. (Thanks to Artie Prendergrast-Smith for this.)  
I should also note that my original movitation for formulating the above argument was similar to that of Keith: the usual proof of the existence of minimal primes in Noetherian rings uses Zorn's lemma and has a non-constructive feel, whereas the above reformulation feels more concrete.