Terminology: Algebras where long strings of products are 0?

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 $i_j in lbrace 1, ldots, krbrace$) 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 $A_{+} = bigoplus_{i=2}^{infty} 2mathbb{Z}/2^imathbb{Z}$.)

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.