Research Article
Year 2024,
Early Access, 1 - 12
Abstract
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.
Year 2024,
Early Access, 1 - 12
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.
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.
There are 9 citations in total.
Details
| Primary Language | English |
|---|---|
| Subjects | Algebra and Number Theory |
| Journal Section | Articles |
| Authors | |
| Early Pub Date | February 17, 2024 |
| Publication Date | |
| Submission Date | October 19, 2023 |
| Acceptance Date | January 6, 2024 |
| Published in Issue | Year 2024 Early Access |