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I'd like a name for an augmented algebra $A = \langle 1\rangle \oplus A_+$ for which there is an $N$ so that any product of more than $N$ elements in the augmentation ideal is $0$, i.e., $(A_+)^N = 0$. Is there a name? It seems related to nilpotence, and it implies that all elements in $A_+$ are nilpotent, but is stronger than that. There is a uniform bound on the degree of nilpotence, but that's not enough either, as the example of the exterior algebra in infinitely many variables over $\mathbb{Z}/2$ shows.

MathWorld defines a nilpotent algebra or nilalgebra to be one where every element is nilpotent. (They are therefore not considering unital algebras, contrary to an earlier discussion here.) Is this standard? Is there a better term?

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up vote 6 down vote accepted

I believe the terminology you want is that "the augmentation ideal is locally nilpotent." See, e.g. http://planetmath.org/encyclopedia/NilAndNilpotentIdeals.html, although there are surely places in actual literature where this is used. I think an ideal being nilpotent means that the powers of the ideal are eventually zero (and is stronger than just requiring it to be a nil-ideal, meaning it's comprised of individually nilpotent elements). So a nilpotent ideal is one where there is a uniform bound, i.e. I^N=0 for some sufficiently large N. Locally nilpotent means that for any finitely generated subalgebra there exists such an N.

Is that the notation you want? I don't know a name for an augmented algebra whose ideal is locally nilpotent other than just that.

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That's good, thanks, although it's a mouthful. The added "locally" condition is not what I asked for. I'd just like to confirm that PlanetMath is contradicting MathWorld on what a nilpotent ring is, and that PlanetMath is right? –  Dylan Thurston Apr 2 '10 at 19:09
Right, sorry. I read that you said you had a uniform bound, but then forgot it. =] –  David Jordan Apr 2 '10 at 20:57
Yes, to confirm, the PlanetMath terminology is standard, while the MathWorld terminology is bad. –  Manny Reyes Apr 2 '10 at 21:52
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Just to confirm the accepted answer, I link here relevant definitions from Springer Online Encyclopaedia of Mathematics (definitely more reliable source than MathWorld and even PlanetMath...):

Nilpotent algebra

Locally nilpotent algebra

Nil algebra

So the most appropriate thing to say would be "the augmentation ideal is nilpotent". This terminology is very standard.

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It seems to me that Jordan and Dotsenko are giving different answers from one another, and I agree with Dotsenko’s.  The condition Thurston has stated is the definition of $A_{+}$ being nilpotent.  “Locally nilpotent” is a weaker condition.  There are many examples of nonunital rings $A_{+}$ that are locally nilpotent (meaning for any finite set $a_1, \ldots, a_k \in A_{+}$ there exists $n \in \mathbb{N}$ such that $a_{i_1} \cdots a_{i_n} = 0$ provided every $i_j \in \lbrace 1, \ldots, k\rbrace$) but not nilpotent (meaning there exists $n \in \mathbb{N}$ such that $a_1 \cdots a_n = 0$ provided every $a_i \in A_{+}$, which is the condition Thurston stated).  Even better: two nice (and quite different) examples of a locally nilpotent prime nonunital ring can be found in E. I. Zelmanov, “An example of a finitely generated primitive ring,” Sibirsk. Mat. Zh. 20 (1979), no. 2, 423, 461, and J. Ram, “On the semisimplicity of skew polynomial rings,” Proc. Amer. Math. Soc. 90 (1984), no. 3, 347–351.  (Of course, if one merely wants an example where $A_{+}$ is locally nilpotent but not nilpotent—and so does not satisfy Thurston’s condition—one could take something like $A_{+} = \bigoplus_{i=2}^{\infty} 2\mathbb{Z}/2^i\mathbb{Z}$.)

N.B. Mathematical Reviews incorrectly lists the title of Zelmanov’s paper as “An example of a finitely generated primary ring.”  It’s listed correctly in Zentralblatt.  Possibly the problem lies in the translation from the original Russian; the condition primitive in the English translation of the paper (Siberian Math. J. 20 (1979), no. 2, 303–304) is what we would today call prime.

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Thanks for the comments! I hadn't thought about that distinction. Local nilpotence seems like a rather odd condition; I don't yet see where it would be useful. –  Dylan Thurston Jun 30 '10 at 23:59
I'm happy to hear this was of some small help. My apologies for the lengthy delay in replying to your message. I'm not sure how persuasive a justification you'll find this, but local nilpotence of algebras is connected with the Burnside Problem for groups, asking whether a finitely generated periodic group need be finite. The first counterexample was an immediate consequence of E. S. Golod's example of a finitely generated nil but not nilpotent algebra (thus in between nil and locally nilpotent). E. Zelmanov discusses local nilpotence of algebras in the context of his solution to the ... –  Greg Marks Nov 1 '10 at 21:40
... Restricted Burside Problem in his 1992 monograph <i>Nil Rings and Periodic Groups</i>. Sample result: C. Procesi's theorem that a periodic group contained in the unit group of a polynomial identity algebra must be locally finite. –  Greg Marks Nov 1 '10 at 21:43
(I guess jsMath gets rendered in the comments sections, but HTML markup doesn't. Weird.) –  Greg Marks Nov 1 '10 at 21:49
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