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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?

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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://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:

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

http://math.uga.edu/~pete/Motzkin49.pdf

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

http://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).

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  • $\begingroup$ Aha! Thanks. I kept trying a direct approach given $a\in R$, or looking at the set of nonzero nonunits that did not have a factorization. $\endgroup$ – Arturo Magidin Feb 15 '10 at 18:58
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    $\begingroup$ Somewhat more seriously, it's safest to only ask one question at a time. The fact that my answer appears and answers your first question will make it harder for people to see that your second question remains unanswered. If you're not getting as much action on that as you'd like, consider asking it again as a separate question, say tomorrow. $\endgroup$ – Pete L. Clark Feb 15 '10 at 19:19
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    $\begingroup$ I haven't seen an ideal theoretic characterization of Euclidean rings; I would even be surprised if there is one, but if there is, then it probably has to start with Motzkin's observation that if a ring is Euclidean for some norm, then it is Euclidean with respect to the minimal Euclidean norm. $\endgroup$ – Franz Lemmermeyer Feb 15 '10 at 19:33
  • $\begingroup$ @FL: Agreed. There is, though, a way to express Euclideanity without making reference to a norm: Motzkin's work accomplishes this. $\endgroup$ – Pete L. Clark Feb 15 '10 at 19:39
  • $\begingroup$ @PC: I thought I had voted it up. I just voted it up and it accepted it, so I guess it didn't take. And thanks for the advice! $\endgroup$ – Arturo Magidin Feb 15 '10 at 21:44
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Although Pete Clark's answer is great, I thought I'd post a partial answer that addresses the UFD question in a different direction. My favorite ideal-theoretic characterization of UFDs is that a domain $R$ is a UFD if and only if every $t$-closed ideal of $R$ is principal. The $t$-closure operation on the fractional ideals of a domain $R$ is given by $$t: I \longmapsto I^t = \bigcup\{(J^{-1})^{-1}: J \subseteq I \mbox{ is a finitely generated ideal of } R\},$$ where $J^{-1} = (R :_{Q(R)} J)$ and $Q(R)$ is the quotient field of $R$. The $t$-closure operation is a useful closure operation on the fractional ideals of a domain $R$. Krull introduced such $'$-Operations in the 1930s in his book Idealtheorie and some subsequent papers. Today we call them star operations.

A UFD is equivalently a Krull domain with trivial divisor class group. A Krull domain is equivalently an integral domain $R$ such that $(II^{-1})^t = R$ for every nonzero ideal $I$ of $R$. It follows from these two well known results of multiplicative ideal theory that a UFD is equivalently a domain in which every ideal $I$ such that $I^t = I$ is principal. I like this characterization because it doesn't mention anything about principal prime ideals or even prime ideals. After all, it is easy to see that a domain $R$ is a UFD if and only if every nonzero principal ideal is a product of principal prime ideals. If you're allowed to mention principal prime ideals in the characterization, then such characterizations are easy to come by.

I would note that the notion of a principal ideal doesn't have a "purely" ideal-theoretic description, since they can't be recovered from the ideal lattice alone. In particular, in my view even the notion of a PID does not have a purely ideal-theoretic description. However, the notion of a Dedekind domain does: a domain $R$ is a Dedekind domain if and only if $II^{-1} = R$ for every nonzero ideal $I$ of $R$. This all hinges on what you mean by ``ideal-theoretic.'' To me it means, can the notion be defined in terms of the ordered monoid of all ideals, or fractional ideals, of the ring. For example, the finitely generated ideals of a commutative ring $R$ have a purely ideal-theoretic description, namely, as the compact elements of the poset of all ideals of $R$.

By the way, a PID is a equivalently a Dedekind domain with trivial ideal class group, and this is generalized by the fact mentioned earlier that a UFD is equivalently a Krull domain with trivial divisor class group.

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