Pro-finite p-adic Lie algebras

Let p be a prime number. A finite nilpotent Lie ring of characteristic a power of p is called finite-p. A pro-p Lie ring is an inverse limit of finite-p Lie rings. Pro-p Lie rings play a role in Lie theory similar to that played by pro-p groups in group theory. Every pro-p Lie ring admits the structure of a Lie algebra over the p-adic integers; furthermore, every p-adic Lie algebra of finite rank as a p-adic module has an open pro-p subalgebra. We make a detailed study of pro-p Lie rings in terms of various properties, including their topology, Prüfer rank, subring growth, and p-adic module structure. In particular, we prove the equivalence of the following conditions for a finitely generated pro-p Lie ring L: L has finite Prüfer rank; L is isomorphic to a closed subring of for some p-adic module V of finite rank; and, for sufficiently large n, the Lie -subalgebra is not an open section of L. By reducing to the pro-p Lie ring case, we also prove that all Engelian pro-finite Lie rings are locally nilpotent. This is a Lie theoretic analogue of Zelmanovʹs theorem which states that every periodic pro-p group is locally finite.