# Free centre-by-nilpotent-by-abelian Lie rings of rank 2

Alexandrou, Maria and Stöhr, Ralph (2014) Free centre-by-nilpotent-by-abelian Lie rings of rank 2. [MIMS Preprint]

We study the free Lie ring of rank $2$ in the variety of all centre-by-nilpotent-by-abelian Lie rings of derived length $3$. This is the quotient $L/([\gamma_c(L'),L]+L''')$ with $c\geqslant 2$ where $L$ is the free Lie ring of rank $2$, $\gamma_c(L')$ is the $c$-th term of the lower central series of the derived ideal $L'$ of $L$, and $L'''$ is the third term of the derived series of $L$. We show that the quotient $\gamma_c(L')+L'''/[\gamma_c(L'),L]+L'''$ is a direct sum of a free abelian group and a torsion group of exponent $c$. We exhibit an explicit generating set for the torsion subgroup.