A Remark on Ado's Theorem for Principal Ideal Domains

Ado's Theorem had been extended to principal ideal domains independently by Churkin and Weigel.They proved that if R is a principal ideal domain of characteristic zero and L is a Lie algebra over R which is also a free R -module of finite rank, then L admits a finite faithful Lie algebra representation over R .We present a quantitative proof of this result, providing explicit bounds on the degree of the Lie algebra representations in terms of the rank as a free module.To achieve it, we generalise an established embedding theorem for complex Lie algebras: any Lie algebra as above embeds in a larger Lie algebra that decomposes as the direct sum of its nilpotent radical and another subalgebra.