RINGS SATISFYING GENERALIZED ENGEL CONDITIONS

Abstract

Let R be an associative ring and let x, y ∈ R. Define the generalized commutators as follows: [x, 0y] = x and [x, ky] = [x, k-1y]y - y[x, k-1y](k = 1, 2, …). In this paper we study some generalized Engel rings, i.e. [Formula: see text]-rings (satisfying [xm(x, y), k(x, y)y] = 0), [Formula: see text]-rings (satisfying [xm(x, y), k(x, y)yn(x, y)] = 0) and [Formula: see text]-rings (satisfying [xm(x, y), k(x, y)yn(x, y)]r(x, y) = 0). Among other results, it is proved that every Artinian [Formula: see text]-ring is strictly Lie-nilpotent. Also, we show that in each of the following cases R has nil commutator ideal: (1) if R is a [Formula: see text]-ring with unity and k, n independent of y; (2) if R is a locally bounded [Formula: see text]-ring (defined below); (3) if R is an algebraic algebra over a field in which R* is a bounded Engel group or a soluble group.