Abstract

We study the variety of algebras A over a field of characteristic ≠ 2, 3, 5 satisfying the identities xy=yx and β ((xx)y)x-((yx)x)x + γ ((xx)x)y-((yx)x)x=0, where β, γ are scalars. We do not assume power-associativity. We prove that if A admits a non-degenerate trace form, then A is a Jordan algebra. We also prove that if A is finite-dimensional and solvable, then it is nilpotent. We find three conditions, any of which implies that a finite-dimensional right-nilalgebra A is nilpotent. © 2008 Academy of Mathematics and Systems Science, Chinese Academy of Sciences, and Suzhou University.