On a Wedderburn principal theorem for the flexible algebras

Abstract

A strictly power-associative algebra A over a field K is said to have a Wedderburn decomposition if there is a subalgebra S of A such that A = S + N A = S + N , where N is the nil radical of A, and S = A − N S = A - N . A Wedderburn principal theorem for a class of algebras is a theorem which asserts that the algebras A, in the class, with A − N A - N separable have Wedderburn decompositions. It is known that there is no such theorem for the class of noncommutative Jordan algebras. A partial result in this direction is the following theorem. Theorem. Let A be a strictly power-associative, flexible algebra over a field F with characteristic not 2 or 3, with A − N A - N separable and such that A = A 1 ⊕ A 2 ⊕ ⋯ ⊕ A n A = {A_1} \oplus {A_2} \oplus \cdots \oplus {A_n} where each A i {A_i} . has A i − N i {A_i} - {N_i} simple and has more than two pairwise orthogonal idempotents. Then A = S + N A = S + N where S is a subalgebra of A.