An interconnection between pre-Lie rings, braces and associative rings

Abstract

Let A be a brace of cardinality $p^n$ for some prime number p. Denote $ann\left(p^i\right)=\{a\in A:p^{i}a=0\}$. Suppose that for $i=1,2,\dots$ and all $a,b\in A$ we have$a⁎(a⁎(\cdots ⁎a⁎b\left)\right)\in pA,a⁎(a⁎(\cdots ⁎a⁎ann\left(p^i\right)\left)\right)\in ann\left(p^{i-1}\right)$ where a appears less than $\frac{p-1}{4}$ times in this expression. Let k be such that $p^{k(p-1)}A=0$. It is shown that the brace $A/ann\left(p^{4k}\right)$ is obtained from a left nilpotent pre-Lie ring by a formula which depends only on the additive group of brace A. We also obtain some applications of this result.