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 $A = S + N$, where N is the nil radical of A, and $S = A - N$. A Wedderburn principal theorem for a class of algebras is a theorem which asserts that the algebras A, in the class, with $A - N$ 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 $A - N$ separable and such that $A = {A_1} \oplus {A_2} \oplus \cdots \oplus {A_n}$ where each ${A_i}$. has ${A_i} - {N_i}$ simple and has more than two pairwise orthogonal idempotents. Then $A = S + N$ where S is a subalgebra of A.