What are Noether's "Fünfaxioms" for Dedekind domains?

Question

While reading the following article

Fuchs, Ladislaus. "Über die Ideale arithmetischer ringe." Commentarii Mathematici Helvetici 23.1 (1949): 334-341.

I ran across this allusion which I'm unable to clarify:

... alle Noetherschen Fünfaxioms-Ringe immer $A$-Ringe sind.

I know little of German but I can get along with machine translation. The best I can make out is "all Noetherian Five-axiom rings are arithmetic rings."

But what's a "Fünfaxioms-Ringe"? Google gives me nothing... Perhaps it is very archaic or I simply don't know something obvious that a fluent German speaker knows.

Can anyone help clear this up?

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Huh naturally only after I posted did I think to attempt "five axiom ring" in English in google, and it got one hit, a biography of Emmy Noether, which says

Furthermore, E. Noether established necessary and sufficient conditions for every ideal to be a product of powers of prime ideals (five-axiom-ring, or Dedekind ring). By eliminating individual axioms...

Switching to google "Noether five axiom" yielded another book by J.S. Milne with a clearer claim:

Emmy Noether re-examined Kummer’s work more abstractly, and named the integral domains for which his methods applied “five-axiom rings”. found here

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So now it is clearer that it means "Dedekind ring." Now my question is: what are the five axioms? (There's no mention of Noether specifically on the wiki page for Dedekind domains.)

Answer 1

I'm German and have never come across this term; I believe it is rather archaic. I'm no historian, so take all my interpretation of this text with a grain of salt. The origin of the term appears to be Emmy Noether's Abstrakter Aufbau der Idealtheorie in algebraischen Zahl- und Funktionenkörpern . The introduction reads

Im folgenden wird eine abstrakte Charakterisierung all derjenigen Ringe gegeben, deren Idealtheorie übereinstimmt mit der Idealtheorie aller ganzen Größen des algebraischen Zahlkörpers — deren Ideale sich also eindeutig als Potenzprodukte von Primidealen darstellen lassen.

In rough translation,

In the following, the rings whose theory of ideals agrees with the theory of ideals in the ring of integers of an algebraic number field — that is, whose ideals can be uniquely written as products of powers of prime ideals — will be abstractly characterized.

In modern language, this means Dedekind rings. Note that rings are specified to be commutative. Then, she lists the five axioms.

I. Teilerkettensatz: Jede Kette von Idealen, bei der jedes Ideal ein echter Teiler des vorangehenden ist, bricht im Endlichen ab. [...]

In rough translation,

I. Theorem on chains of divisors: Every chain of ideals in which each ideal is a proper divisor of the preceding one terminates in the finite.

In modern language, the ring is Noetherian. Further,

II. Vielfachen-Kettensatz modulo jedem vom Nullideal verschiedenen Ideal: Jede Kette von Idealen — die sämtlich Teiler eines festen, vom Nullideal verschiedenen Ideals sind —, bei der jedes Ideal ein echtes Vielfaches des vorangehenden ist, bricht im Endlichen ab.

In rough translation,

II: Theorem on chains of multiples modulo a fixed ideal different from the zero ideal: Every chain of ideals — each of which dividing a fixed ideal different from the zero ideal —, in which each ideal is a proper multiple of the preceding one terminates in the finite.

In modern language, each proper quotient of the ring is Artinian. Further,

III: Existenz des Einheitselementes der Multiplikation.

In rough translation,

III: Existence of a unit element for multiplication.

This is self-explanatory, though surprising. It seems to suggest that "ring" in this time period meant not necessarily unital rings. Further,

IV: Ring ohne Nullteiler.

In rough translation,

IV: Ring without zero divisors.

This is self-explanatory. Lastly,

V: Ganze Abgeschlossenheit im Quotientenkörper: Jedes Element des Quotientenkörpers, das ganz in bezug auf den Ring ist, gehört dem Ring an.

In rough translation,

V: Integral closure in the quotient field: Every element of the quotient field, which is integral relative to the ring, belongs to the ring.

In modern language, the ring is integrally closed.

Thus, after interpretation and recasting everything in modern language, the crux appears to be the characterization of Dedekind rings as precisely the integrally closed domains that are Noetherian and each proper quotient of which is Artinian. This characterization is, of course, well-known to this day.