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For a Lie ring L over the ring of integers, we compare its lower central series {γn(L)}n≥1 and its dimension series {δn(L)}n≥1 defined by setting δn(L) = L ∩ ϖn(L), where ϖ(L) is the augmentation ideal of the universal enveloping algebra of L. While γn(L) ⊆ δn(L) for all n ≥ 1, the two series can differ. In this paper, it is proved that if L is a metabelian Lie ring, then 2δn(L) ⊆ γn(L), and [δn(L),L] = γn+1(L), for all n ≥ 1.

Dimension quotients of metabelian Lie rings