On pre-Lie rings related to some non-Lazard braces

Abstract

Let A be a brace of cardinality $p^{n}$ for some prime number $p$. Suppose that either (i) the additive group of brace $A$ has rank smaller than $p-3$, or (ii) $A^{\frac {p-1}2}\subseteq pA$ or (iii) $p^{i}A$ is an ideal in in $A$ for each $i$. It is shown that there is a pre-Lie ring associated to brace $A$. The left nilpotency index of this pre-Lie ring can be arbitrarily large. Let $A$ be a brace of cardinality $p^{n}$ for some prime number $p$. Denote $ann(p^{i})=\{a\in A: p^{i}a=0\}$. Suppose that for $i=1,2,\ldots $ and all $a,b\in A$ we have \[a*(a*(\cdots *a*b))\in pA, a*(a*(\cdots *a*ann(p^{i})))\in ann(p^{i-1})\] 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 $A/ann(p^{4k})$ 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.