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?



Characterizations of UFD and Euclidean domain by ideal-theoretic conditions
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](http://alpha.math.uga.edu/%7Epete/factorization.pdf)


Regarding your second question, there has been some work on understanding Euclidean domains from more intrinsic perspectives. Two fundamental articles are:



> 
> Motzkin, Th. The Euclidean algorithm. Bull. Amer. Math. Soc. 55, (1949). 1142--1146.
> 
> 
> 


[http://alpha.math.uga.edu/~pete/Motzkin49.pdf](http://alpha.math.uga.edu/%7Epete/Motzkin49.pdf)



> 
> Samuel, Pierre About Euclidean rings. J. Algebra 19 1971 282--301.
> 
> 
> 


[http://alpha.math.uga.edu/~pete/Samuel-Euclidean.pdf](http://alpha.math.uga.edu/%7Epete/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).


