Non-Abelian Extension Theory for Lie Rings

In this paper, we establish fundamental results concerning non-abelian extensions of Lie rings. Specifically, we demonstrate one-to-one correspondence between the equivalence classes of non-abelian extensions of Lie ring Formula: see text by Lie ring Formula: see text and the elements of second cohomology group Formula: see text, where Formula: see text acts on Formula: see text. Additionally, we provide necessary and sufficient condition under which such extensions exist, particularly when a specific abstract kernel is involved.