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A Generalization of Commutative and Alternative Rings

Published online by Cambridge University Press:  20 November 2018

Erwin Kleinfeld ,
Margaret Humm Kleinfeld  and
Frank Kosier
Erwin Kleinfeld
Affiliation:
The University of Iowa, Iowa City, Iowa
Margaret Humm Kleinfeld
Affiliation:
The University of Iowa, Iowa City, Iowa
Frank Kosier
Affiliation:
The University of Iowa, Iowa City, Iowa
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In [3] Schafer has defined generalized standard rings as rings satisfying the identities

(1)

(2)

(3)

and observed that these identities imply (y,y, (x, z)) = 0 and if the characteristic is not three, (x, y, x2) = 0. Schafer determined the structure of simple, finite-dimensional generalized standard algebras of characteristic not two or three by showing that they must be either commutative, Jordan, or alternative.

Previously one of us [2] had studied accessible rings, which are defined by the identities (x,y,z) + (z,x,y) – (x,z,y) = 0 and ((w,x), y,z) = 0.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 22 , Issue 2 , 01 April 1970 , pp. 348 - 362
Copyright
Copyright © Canadian Mathematical Society 1970

References

1. Albert, A. A., Power associative rings, Trans. Amer. Math. Soc. 64 (1948), 552–593.Google Scholar
2. Erwin, Kleinfeld, Standard and accessible rings, Can. J. Math. 8 (1956), 335–340.Google Scholar
3. Schafer, R. D., On generalized standard algebras, Proc. Nat. Acad. Sci. U.S.A. 60 (1968) 73–74.Google Scholar