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](https://mathoverflow.net/questions/15107/algebra-unital-associative-algebra-better-terminology) here.) Is this standard? Is there a better term?



Terminology: Algebras where long strings of products are 0?
I believe the terminology you want is that "the augmentation ideal is locally nilpotent." See, e.g. [Link](https://planetmath.org/NilAndNilpotentIdeals), 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.


