Re: [Question] Dedekind Domain

Arturo Magidin <magidin@xxxxxxxxxxxxxxxxx> wrote:
>Taedong Yun <noname@xxxxxxxxxxxxx> wrote:
>>
>> Let R be a domain with the property that every nonzero ideal
>> is a product of maximal ideals. Show that R is Dedekind.
>
> This proof is in Jacobson's "Basic Algebra II", second edition,
> Chapter 10, Section 2, Theorem 10.5. The proof is attributed to
> Zariski and Samuel.

This result is older than Zariski and Samuel's textbook (1958).
For example, see
------------------------------------------------------------------------------
7,360i 09.1X
Matusita, Kameo
Uber ein bewertungstheoretisches Axiomensystem fur die Dedekind-Noethersche
Idealtheorie. (German)
Jap. J. Math. 19 (1944). 97-110.
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A valuation-theoretic criterion is given for an integral domain D to
have the classical Dedekind-Noether ideal theory, that is, for every proper
ideal of D to be uniquely expressible as a product of maximal prime
ideals. A necessary and sufficient condition is: (1) D is the
intersection of a set of discrete valuation rings; (2) each element of D
is a unit in all but a finite number of the valuation rings; (3) given any
two of the valuation rings, there is an element e in D such that e
is a non-unit in one and 1-e is a non-unit in the other. This criterion is
applied to give alternate proofs of several known theorems. It is also shown
that, if every proper ideal in D is the product of prime ideals, then
the primes must be maximal and the decomposition unique.
Reviewed by I. S. Cohen
------------------------------------------------------------------------------
11,413g 09.1X
Cohen, I. S.
Commutative rings with restricted minimum condition.
Duke Math. J. 17, (1950). 27-42.
------------------------------------------------------------------------------
The rings which are dealt with in this paper are commutative and have an
identity element; an ideal in a ring R is called proper if it is different from
both {0} and R . The object of the paper is to study the interrelationship of
various conditions which a ring may satisfy: the maximum condition (Noetherian
rings), the minimum condition, the restricted minimum condition (i.e., the
minimum condition should hold in R/I for every proper ideal I ), the
condition of being a Dedekind ring, the condition of being of finite rank
( R is of rank k if every ideal has a set of k generators).

Theorem 1 states that the restricted minimum condition is satisfied in R if and
only if R is Noetherian and every proper prime ideal is maximal; this contains
the result due to Akizuki [Proc. Phys.-Math. Soc. Japan (3) 17, 337-345 (1935)]
to the effect that the restricted minimum condition implies the maximal
condition.

Theorem 3 is as follows: let R be a restricted minimum integral domain; let S
be an integral domain containing R whose quotient field is finite over that of
R and which is integral over R ; then S is a restricted minimum ring, and, more
precisely, if I is a proper ideal of S , then S/I has a composition series when
considered as an R-module. Various special cases of this result have been
proved before; a result whose equivalence to theorem 3 is established in the
present paper has been announced without proof by H. Grell [Ber. Math.-Tagung
T?bingen 1946, p. 67 (1947); these Rev. 9,5].

A simple proof is given of the theorem due to Matusita [Jap. J. Math. 19,
97--110 (1944); these Rev. 7,360] to the effect that an integral domain in
which every ideal is a product of prime ideals is a Dedekind ring. The
following characterisations of Dedekind rings are given: if R is a
Noetherian integral domain, then the following conditions are all equivalent:
(1) R is a Dedekind ring;
(2) for every maximal ideal M, the quotient ring R_M is a DVR;
(3) if M is a maximal ideal of R, there is no ideal between M and M^2;
(4) a primary ideal belonging to a maximal ideal is a product of prime ideals;
(5) the set of primary ideals belonging to a maximal ideal is totally ordered;
(6) for any three ideals in R, A /\(B + C) = (A /\B) + (A /\C);
(7) for any three ideals of R, A : (B /\C) = (A : B) + (A : C).

It is proved that a local domain (Noetherian integral domain in which the
non-units form an ideal) is of finite rank if and only if it satisfies the
restricted minimum condition [the sufficiency had been proved by Akizuki, Jap.
J. Math. 14, 85-102 (1938)]. Moreover it is established that for a Noetherian
domain R to be of finite rank, it is necessary and sufficient that the ranks
of the rings R_P for all prime ideals P be bounded; if k is an upper bound for
these ranks, then R is of rank k+1. Some applications are given to modules
of finite rank over a ring of finite rank.
Reviewed by C. Chevalley

--Bill Dubuque
.

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