On the Wedderburn principal theorem

Abstract

Proof.We will prove the first part by induction on t.When r=l, u=Uy.In addition, [uk] = [u]k = [u] so u is not nilpotent.Hence, the associative algebra of all polynomials in u, denoted by F[u], is not nilpotent and thus contains an idempotent e=f(u) for fix) a polynomial.Then [e] = [/()] = a[u] where a=f(l) is in F. Thus a[w] = [e] = [e]2 = c72[w]2 = c72[w].Since e is not in M, a[u]^0 and a=l.Let w = Uy + u2 for r^2.Then u=w+u3+ +ut for pairwise orthogonal idempotents [w], [u3],..., [ut].By the inductive hypothesis, there exists an idempotent e and pairwise orthogonal idempotents/ e3,...,et in Ae(l) such that