Idempotents and zero divisors in commutative algebras satisfying an identity of degree four

Abstract

We study commutative algebras satisfying the identity $ ((wx)y)z+((wy)z)x+((wz)x)y-((wy)x)z- ((wx)z)y-((wz)y)x = 0. $ We assume characteristic of the field $\neq 2,3.$ We prove that given any $\lambda \in F,$ there exists a commutative algebra with idempotent $e,$ which satisfies the identity, and has $\lambda $ as an eigen value of the multiplication operator $L_e$. For algebras with idempotent, the containment relations for the product of the eigen spaces are not as precise as they are for Jordan or power-associative algebras. A great part of this paper is calculating these containment relations.

Keywords

• Commutative algebra
• degree four identity
• idempotent element
• Peirce decomposition

References

1. A. A. Albert, Power-associative rings, Trans. Amer. Math. Soc., 64 (1948), 552-593.
2. L. Carini, I. R. Hentzel and G. M. Piacentini Cattaneo, Degree four identities not implied by commutativity, Comm. Algebra, 16(2) (1988), 339-356.
3. I. Correa and I. R. Hentzel, Commutative non associative nil algebras satisfying an identity of degree four, Preprint (2023).
4. M. Gerstenhaber, On nilalgebras and linear varieties of nilpotent matrices II, Duke Math. J., 27 (1960), 21-31.
5. D. P. Jacobs, S. V. Muddana and A. J. Offutt, A computer algebra system for nonassociative identities, Hadronic mechanics and nonpotential interactions, Nova Science Publishers, Inc., Commack, NY, Part 1 (Cedar Falls, IA, 1990) (1992), 185-195.
6. D. P. Jacobs, D. Lee, S. V. Muddana, A. J. Offutt, K. Prabhu and T. Whiteley, Albert's User Guide, Version 3.0., Department of Computer Science, Clemson University, 1996.
7. J. M. Osborn, Commutative non-associative algebras and identities of degree four, Canadian J. Math., 20 (1968), 769-794.
8. C. Rojas-Bruna, Trace forms and ideals on commutative algebras satisfying an identity of degree four, Rocky Mountain J. Math., 43(4) (2013), 1325-1336.

9. R. D. Schafer, An Introduction to Nonassociative Algebras, Pure Appl. Math., 22, Academic Press, New York-London, 1966.