Schur’s theorem states that for a group \(G\), if the central factor \(G/Z(G)\) is finite, then so is the derived subgroup \(G'\). Also, Baer’s theorem states that if for a natural number \(i\) the quotient \(G/Z_{i}(G)\) is finite, then so is \(\gamma_{i+1}(G)\), where \(Z_{i}(G)\) and \(\gamma_{i+1}(G)\) are \(i\)-th and \((i+1)\)-st terms of the upper and lower central series of \(G\), respectively. Several kinds of variations of these results, for groups, Lie algebras and other types of algebraic objects, exist in the literature.
In the paper under review, it is discussed Schur-type theorems for Lie and Leibniz algebras. It is proved that for a given finitely generated Leibniz algebra \(\mathfrak{m}\), the commutator \([\mathfrak{m},\mathfrak{m}]\) is finitely generated if and only if for every central extension \(0\to \mathfrak{r}\to \mathfrak{l}\to \mathfrak{m}\to 0\) of Leibniz algebras, the commutator \([\mathfrak{l},\mathfrak{l}]\) is finitely generated. Moreover, it is proved that if \(\mathfrak{m}\) is a finitely generated Lie algebra and \(i\) a natural number, then \(\gamma_{i+1}(\mathfrak{m})\) is finitely generated if and only if for every \(i\)-central extension \(0\to \mathfrak{r}\to \mathfrak{l}\to \mathfrak{m}\to 0\) of Lie algebras, \(\gamma_{i+1}(\mathfrak{l})\) is finitely generated. Finally in the last section, it is discussed similar results for finitely presented Lie algebras.