On the Wedderburn principal theorem for nearly (1,1) algebras

Abstract

A nearly ( 1 , 1 ) (1,1) algebra is a finite dimensional strictly power-associative algebra satisfying the identity ( x , x , y ) = ( x , y , x ) (x,x,y) = (x,y,x) where the associator ( x , y , z ) = ( x y ) z − x ( y z ) (x,y,z) = (xy)z - x(yz) . An algebra A has a Wedderburn decomposition in case A has a subalgebra S ≅ A − N S \cong A - N with A = S + N A = S + N (vector space direct sum) where N denotes the radical (maximal nil ideal) of A. D. J. Rodabaugh has shown that certain classes of nearly ( 1 , 1 ) (1,1) algebras have Wedderburn decompositions. The object of this paper is to expand these classes. The main result is that a nearly ( 1 , 1 ) (1,1) algebra A containing 1 over a splitting field of characteristic not 2 or 3 such that A has no nodal subalgebras has a Wedderburn decomposition.