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

Year 2024, Volume: 36 Issue: 36, 194 - 205, 12.07.2024

Manuel Arenas Ivan Correa Irvin Roy Hentzel Alicia Labra

https://doi.org/10.24330/ieja.1438748

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

• A. A. Albert, Power-associative rings, Trans. Amer. Math. Soc., 64 (1948), 552-593.
• L. Carini, I. R. Hentzel and G. M. Piacentini Cattaneo, Degree four identities not implied by commutativity, Comm. Algebra, 16(2) (1988), 339-356.
• I. Correa and I. R. Hentzel, Commutative non associative nil algebras satisfying an identity of degree four, Preprint (2023).
• M. Gerstenhaber, On nilalgebras and linear varieties of nilpotent matrices II, Duke Math. J., 27 (1960), 21-31.
• 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.
• 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.
• J. M. Osborn, Commutative non-associative algebras and identities of degree four, Canadian J. Math., 20 (1968), 769-794.
• 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.
• R. D. Schafer, An Introduction to Nonassociative Algebras, Pure Appl. Math., 22, Academic Press, New York-London, 1966.