Primitive elements of free Lie $p$-algebras

Abstract

Let $L$ be a finitely generated free Lie $p$-algebra and $\langle a\rangle$ an ideal generated by $a\in L$. It is proved that $L/\langle a \rangle$ is free if and only if $\langle a\rangle$ is primitive (i.e. $a$ belongs to some set of free generators of $L$). Earlier analogues theorems were proved for some objects, for example, for groups, Lie algebras, free algebras and so on.