Abstract
In this paper we study ordinary and restricted enveloping algebras satisfying a polynomial identity. We first show how the Δ-methods of J. Bergen and D. S. Passman ( J. Algebra, in press) can be used to handle U( L) in all characteristics and u( L) if the latter ring is prime. In particular, we offer a simpler proof of the result of Yu. A. Bachturin ( J. Austral. Math. Soc. 18, 1974, 10–21) on ordinary enveloping algebras in characteristic p > 0. Our main theorem asserts that, in general, u( L) is p.i. if and only if L has a restricted subalgebra of finite codimension which is (essentially) commutative. These results are clearly the Lie algebra analogs of the p.i. group ring theorems of M. K. Smith ( J. Algebra 18, 1971, 477–499) and D. S. Passman ( Pacific J. Math. 36, 1971, 467–483).
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