# History of the characterization of commutative Artin rings

When it comes to the world of "classical" (pre-homological) Noetherian commutative algebra, I tend to think of most of the results (Krull's intersection theorem, the principal ideal theorem, etc.) as being ultimiately consequences of the following two "primordial theorems" of the field:

1. The Noether-Lasker theorem, primary decomposition of ideals and modules.
2. The characterization of local commutative Artin rings as zero-dimensional local Noetherian rings (or equivalently finite-length local rings, and local Noetherian rings with nilpotent maximal ideal).

There are two things to note before I ask my question. First of all, I'm referring to what I call "Noetherian commutative algebra." In the more general field of commutative algebra, I'd probably name Nakayama's Lemma and the Cohen-Seidenberg theorems (Lying Over, Incomparability, Going-Up, and Going-Down). Tangentially, I'd be interested to hear which theorems others would give that distinction. Secondly, some might question my choice of 2. here. Although it doesn't appear directly in proofs very often, it is a crucial lemma in the proof of the Principal Ideal Theorem, and I feel it's a sufficiently interesting and substantial statement to deserve its own place.

Here is my question. The Noether-Lasker theorem has a pretty well-known history. (Chess champion Lasker proved it for polynomial and power series rings, and then Noether developed its current form, as described on Wikipedia.) As for "theorem 2," however, I've never heard a account of its origins. Was it first introduced by Krull to prove the Principal Ideal Theorem, or was it known prior? Perhaps Krull's original proof did not even use that lemma, and it was introduced later.

Does anyone have a reference on that history?

• Well, the fact that Artinian rings are Noetherian is the Akizuki-Hopkins theorem (or maybe there are three authors?). After that, the nilpotence of the maximal ideal follows from a combination of the Artin condition and the Nakayama lemma (so that $m^k \neq m^{k+1}$ unless $m^k = 0$, where $m$ is the maximal ideal). Jun 7, 2013 at 1:36
• @Neil: I'm well aware of the proof in the commutative case, although I've never studied noncommutative rings, so I wasn't previously familiar with Akizuki-Hopkins. It appears that Akizuki and Hopkins were contemporaries of Krull, so perhaps this (much more general result) was in fact known to Krull at the time that he used the commutative case. Jun 9, 2013 at 15:14