Dimension Quotients of Metabelian Lie Rings

For a Lie ring $L$ over the ring of integers, we compare its lower central series $\{γ_n(L)\}_{n\geq 1}$ and its dimension series $\{δ_n(L)\}_{n\geq 1}$ defined by setting $δ_n(L)= L\cap \varpi^n(L)$, where $\varpi(L)$ is the augmentation ideal of the universal enveloping algebra of $L$. While $γ_n(L)\subseteqδ_n(L)$ for all $n\geq 1$, the two series can differ. In this paper it is proved that if $L$ is a metabelian Lie ring, then $2δ_n(L)\subseteqγ_n(L)$, and $[δ_n(L),\,L]=γ_{n+1}(L)$, for all $n\geq 1$.

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