Vol. 29, No. 1, 1969

Download this article
Download this article. For screen
For printing
Recent Issues
Vol. 294: 1
Vol. 293: 1  2
Vol. 292: 1  2
Vol. 291: 1  2
Vol. 290: 1  2
Vol. 289: 1  2
Vol. 288: 1  2
Vol. 287: 1  2
Online Archive
The Journal
Editorial Board
Special Issues
Submission Guidelines
Submission Form
Author Index
To Appear
ISSN: 0030-8730
Standard algebras

R. D. Schafer

Vol. 29 (1969), No. 1, 203–223

In 1948 A. A. Albert defined a standard algebra A by the identities (x,y,z) + (z,x,y) (x,z,y) = 0 and

(x,y,wz )+ (w,y,xz)+ (z,y,wx ) = 0.

Standard algebras include all associative algebras and commutative Jordan algebras. The radical R of any finite-dimensional standard algebra A is its maximal nilpotent ideal. It is known that any semisimple standard algebra is a direct sum of simple ideals, and that any simple standard algebra is either associative or a commutative Jordan algebra.

In this paper we study Peirce decompositions and derivations of standard algebras. We prove the Wedderburn principal theorem for standard algebras of characteristic 2 (announced in 1950 by A. J. Penico for characteristic 0): if AR is separable, then A = S + R where S is a subalgebra of A, SAR. For standard algebras of characteristic 0 we prove analogues of the Malcev-Harish-Chandra theorem and the first Whitehead lemma, and we determine when the derivation algebra of A is semisimple.

Mathematical Subject Classification
Primary: 17.40
Received: 24 October 1967
Published: 1 April 1969
R. D. Schafer