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*.