Nilpotent ring

Definition
A non-associative ring (i.e., not necessarily associative, commutative, or unital) $$R$$ is termed a nilpotent ring if it satisfies the following conditions:


 * 1) There exists some constant $$c$$ such that for any choice of (possibly repeated) elements $$x_1,x_2, \dots, x_c,x_{c+1} \in R$$, and any sequence $$a_1,a_2,\dots,a_c,a_{c+1}$$ such that $$a_1 = x_1$$ and $$a_{i+1}$$ is one of the products $$a_i * x_{i+1}$$ or $$x_{i+1}*a_i$$, we must have $$a_{c+1} = 0$$.
 * 2) There exists some constant $$m$$ such that for any choice of (possibly repeated) elements $$x_1,x_2, \dots, x_m \in R$$, any product (regardless of the parenthesization and ordering) that involves each $$x_i$$ exactly once has the value 0.
 * 3) The upper central series of $$R$$ reaches $$R$$ in finitely many steps.

The smallest values $$c$$ and $$m$$ that work are two different measures of nilpotency. We have the inequalities $$c+1 \le m \le 2^c$$. For a Lie ring (specifically a nilpotent Lie ring) or an associative ring (specifically a nilpotent associative ring), we have $$c+1 = m$$.

Weaker properties

 * Stronger than::Solvable ring