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Power-Associative Algebras in which Every Subalgebra is a Left Ideal

Published online by Cambridge University Press:  20 November 2018

D. J. Rodabaugh
D. J. Rodabaugh*
Affiliation:
University of Missouri, Columbia, Missouri
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By an L-algebra we mean a power-associative nonassociative algebra (not necessarily finite-dimensional) over a field F in which every subalgebra generated by a single element is a left ideal. An H-algebra is a power-associative algebra in which every subalgebra is an ideal. The H-algebras were characterized by D. L. Outcalt in [2]. Let Sα be the semigroup with cardinality α such that if x, y ∊ Sα then xy = y. Consider the algebra over a field F with basis Sα. Such an algebra is an L-algebra that is not an H-algebra unless Sα contains only one element. In this paper we will prove that an algebra A over a field F with char. ≠ 2 is an L-algebra if and only if it is either an H-algebra or has a basis Sα where α is the dimension of A.

Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 13 , Issue 2 , June 1970 , pp. 239 - 243
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. Outcalt, D. L., Power-associative algebras in which every subalgebra is an ideal, Pacif. J. Math. 20 (1967), 481-485.Google Scholar
3. Lin, Shao-Xue (Lin Shao-Haueh), On algebras in which every subalgebra is an ideal, Acta Math. Sinica 14 (1694), 532-537 (Chinese); translated as Chinese Math.?Acta 5 (1964), 571-577.Google Scholar