Matrix rings and linear groups over a field of fractions of enveloping algebras and group rings. I
The goal here is to show that the ring \(D_ n\) of \(n\times n\) matrices over the division ring of fractions D of a universal enveloping algebra enjoys properties similar to that of a PI-algebra over a field. Let H be a Lie algebra over the field K, let U(H) be its universal enveloping algebra and let D be the ring of fractions of U(H). The main result is as follows. Theorem A. Suppose that H satisfies one of the three conditions: i. U(H) is an Ore ring and \(\cap^{\infty}_{m=1}H^ m=0\). ii. H is solvable of class 2. iii. H is finite dimensional over K. Then \(D_ n\) satisfies: 1. The Jacobson radical of any finitely generated subring of \(D_ n\) is nilpotent of index \(\leq n.\) 2. If V is any finitely generated subring of \(D_ n\) such that any element of V is a sum of nilpotent elements of V, then \(V^ n=0.\) 3. Any noncentral normal subgroup of \(GL_ n(D)\) contains a noncyclic free subgroup. 4. Any periodic subgroup of \(GL_ n(D)\) is locally finite. 5.If \(char K=0\) then any periodic subgroup of \(GL_ n(D)\) contains an Abelian normal subgroup of index \(\leq \rho(n)\) where \(\rho\) is the Jordan function. It is also shown that if H is solvable with \(\cap^{\infty}_{n=1}H^ n=0\), then any nilpotent subgroup of \(D^*\) is Abelian. Moreover, when \(char K=0,\) then any nilpotent-by-finite subgroup of \(D^*\) is Abelian. Finally the paper considers division rings of fractions of group rings. Let G be a solvable residually torsion free nilpotent group or a torsion free polycyclic-by-finite group. If D is the division ring of fractions of KG, then \(D_ n\) also satisfies the conclusions of Theorem A.
division rings of fractions of group rings
solvable residually torsion free nilpotent group
scientific article; zbMATH DE number 3851318