Damghan University Press Analytical and Numerical Solutions for Nonlinear Equations 3060-785X 6 1 2021 08 01 Upper and Lower Central Series in a Pair of Lie Algebras 25 31 193 10.22128/gadm.2020.381.1034 EN Fatemeh Pazandeh Sh. School of Mathematics and Computer sciences, Damghan University, Damghan, Iran Asadollah Faramarzi Salles Damghan University Journal Article 2020 10 10 The Baer's theorem in the termes of the Lie algebras states that for a Lie algebra $L$ the finiteness of $\mathrm{dim}(L/Z_i(L))$ implies the finiteness of $\mathrm{dim}(\gamma_{i+1}(L))$. Let $(N,L)$ denote a pair of Lie algebras, where $N$ is an ideal of $L$, and $d_i=d_i(L)$ denote the minimal number of generators of $L/Z_i(N, L)$. In this paper we shall consider the pair $(N, L)$ and show that if $d_n$ is finite then the converse of Baer's theorem is true. In fact we shall show that if $d_n$ and $\mathrm{dim}(\gamma_{i+1}(N, L))$ are finite, where $i\geq n$, then $N/Z_i(N, L))$ is finite. In particular, we shall provide an upper bound as following, $$\mathrm{dim}(\frac{N}{Z_i(N, L)}) \leq ((d_n)^nd_nd_{n+1}\ldots d_{i-1})\mathrm{dim}(\gamma_{i+1}(N, L))$$$$\leq (d_n)^i(\mathrm{dim}\gamma_{i+1}(N, L)).$$ for all non negative integers i. https://ansne.du.ac.ir/article_193_2614a731cf7843d6fe5c0df5d760c0bb.pdf