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Weakly Semi-Simple Finite-Dimensional Algebras

Published online by Cambridge University Press:  20 November 2018

W. Edwin Clark
W. Edwin Clark*
Affiliation:
University of Florida
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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.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 18 , 1966 , pp. 433 - 442
Copyright
Copyright © Canadian Mathematical Society 1966

References

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4. Schwarz, S., Zur Theorie der Halbgruppen, Sbornik Prac Prirodovedeckej Fakulty Slovenskej Univerzity v Bratislave, no. 6 (1943).Google Scholar
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