Weakly Semi-Simple Finite-Dimensional Algebras

Abstract

Let A be a finite-dimensional (associative) algebra over an arbitrary field F. We shall say that a semi-group S is a translate of A if there exist an algebra B over F and an epimorphism ϕ: B → F such that A = 0→-1 and S = 1→-1. It is shown in (2) that any such semi-group S has a kernel (defined below) that is completely simple in the sense of Rees. Following Stefan Schwarz (4), we define the radical R(S) of S to be the union of all ideals I of S such that some power In of I lies in the kernel K of S. First we prove that the radical of a translate of A is a translate of the radical of A. It follows that A is nilpotent if and only if it has a translate S such that R (S) = S.