Sunday, 22 February 2009

ra.rings and algebras - Characterizations of UFD and Euclidean domain by ideal-theoretic conditions

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 1neq0 and no zero divisors).



One direction is easy: if R is a UFD, and P is a nonzero prime ideal, let ainP, aneq0. Then factor a into irreducibles, a=c1cdotscm. Since P is a prime ideal in a commutative ring, it is completely prime so there is an i such that ciinP, and therefore, (ci)subseteqP. Since ci is a prime element (because R is a UFD), the ideal (ci) 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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