This questions is inspired by an exercise in Hungerford that I have only partially solved. The exercise reads: "A domain is a UFD if and only if every nonzero prime ideal contains a nonzero principal ideal that is prime." (For Hungerford, 'domain' means commutative ring with $1\neq 0$ and no zero divisors).
One direction is easy: if $R$ is a UFD, and $P$ is a nonzero prime ideal, let $a\in P$, $a\neq 0$. Then factor $a$ into irreducibles, $a = c_1\cdots c_m$. Since $P$ is a prime ideal in a commutative ring, it is completely prime so there is an $i$ such that $c_i\in P$, and therefore, $(c_i)\subseteq P$. Since $c_i$ is a prime element (because $R$ is a UFD), the ideal $(c_i)$ is prime.
I confess I am having trouble with the converse, and will appreciate any hints.
But on that same vein, I started wondering if there was a similar "ideal theoretic" condition that describes Euclidean domains. Other classes of domains have ideal theoretic definitions: PID is obvious, of course, but less obvious perhaps are that GCD domains can be defined by ideal-theoretic conditions (given any two principal ideals $(a)$ and $(b)$, there is a least principal ideal $(d)$ that contains $(a)$ and $(b)$, least among all principal ideals containing $(a)$ and $(b)$), as can Bezout domains (every finitely generated ideal is principal). Does anyone know if there is an ideal theoretic definition for Eucldean domains?
Best Answer
Dear Arturo,
The exercise in question is actually a theorem of Kaplansky. It appears as Theorem 5 on page 4 of his Commutative Rings. [I was not able to tell easily whether the result appears for the first time in this book.] The proof is reproduced in Section 10 of an expository article I have written [but probably not yet finished] on factorization in integral domains:
http://alpha.math.uga.edu/~pete/factorization.pdf
Regarding your second question, there has been some work on understanding Euclidean domains from more intrinsic perspectives. Two fundamental articles are:
http://alpha.math.uga.edu/~pete/Motzkin49.pdf
http://alpha.math.uga.edu/~pete/Samuel-Euclidean.pdf
I have not had the chance to digest these papers, so I'm not sure if they directly answer your question (maybe not, but I think they will be helpful).