Title: Lie structure of prime rings of characteristic 2
Abstract: In this paper the Lie structure of prime rings of characteristic 2 is discussed.Results on Lie ideals are obtained.These results are then applied to the group of units of the ring, and also to Lie ideals of the symmetric elements when the ring has an involution.This work extends recent results of I. N. Herstein, C. Lanski and T. S. Erickson on prime rings whose characteristic is not 2, and results of S. Montgomery on simple rings of characteristic 2. LEMMA 2. Suppose 2R -0 and U is a commutative Lie ideal of R. Then u 2 eZ for all ue U. Proof.Let ue U, xe R. Then ux + xue Uso uxu + u 2 x = xu 2 + uxu.Hence u 2 e Z. LEMMA 3. Let R be prime and I a nonzero ideal of R. If [x, I] = 0, 117 118 CHARLES LANSKI AND SUSAN MONTGOMERY then xe Z. Proof.If reR, then for y e /, ry e I and so 0 = [x, ry] = [x, r]y.Thus [x, R]I -0.Since R is prime [x, R] = 0, so xe Z.THEOREM 4. Let R be prime and U a commutative Lie ideal of R. Then Ua Z unless char R -2, Z Φ 0 and RZ~γ is a simple ring ^-dimensional over its center.