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Vol. 51 No. 3–4 (2025)
Serdica Mathematical Journal

Articles

Polynomial identities of finite prime universal algebras

https://doi.org/10.55630/serdica.2025.51.283-298

Published 23-12-2025

  • Yuri Bahturin
    • ,
  • Daniela Martinez Correa
    • ,
  • Diogo Diniz
    • ,
  • Felipe Yasumura

Yuri Bahturin
Memorial University of Newfoundland

Daniela Martinez Correa
Universidade de São Paulo

Diogo Diniz
Federal University of Campina Grande

Felipe Yasumura
Universidade de São Paulo

Abstract

We prove that two finite prime \(\Omega\)-algebras defined over the same unital commutative ring and satisfying the same set of polynomial identities are isomorphic.

2020 Mathematics Subject Classification:

17A42, 16R99

Keywords

Polynomial identities, Universal algebras, Prime algebras

Full text

Author Details

Yuri Bahturin

Department of Mathematics and Statistics
Memorial University of Newfoundland
St. John's, NL, A1C5S7, Canada
e-mail: bahturin@mun.ca

Daniela Martinez Correa

Department of Mathematics
Instituto de Matemática, Estatística e Ciência da Computação
Universidade de São Paulo, SP, Brazil
e-mail: danielam.correa@ime.usp.br

Diogo Diniz

Unidade Acadêmica de Matemática
Universidade Federal de Campina Grande
Campina Grande, PB, 58429-970, Brazil
e-mail: diogo@mat.ufcg.edu.br

Felipe Yasumura

Department of Mathematics
Instituto de Matemática, Estatística e Ciência da Computação
Universidade de São Paulo, SP, Brazil
e-mail: fyyasumura@ime.usp.br

References

  1. E. Aljadeff, D. Haile. Simple G-graded algebras and their polynomial identities. Trans. Amer. Math. Soc. 366, 4 (2014), 1749–1771.
  2. Y. Bahturin. Identical relations in Lie algebras. De Gruyter Exp. Math., vol. 68. Berlin, De Gruyter, 2021.
  3. Ju. A. Bahturin, A. Ju. Ol’shanskiĭ. Identical relations in finite Lie rings. Mat. Sb. (N.S.) 96(138), 4 (1975), 543–559, 645 (in Russian).
  4. Y. Bahturin, F. Yasumura. Distinguishing simple algebras by means of polynomial identities. S˜ao Paulo J. Math. Sci. 13, 1 (2019), 39–72.
  5. Y. Bahturin, F. Yasumura. Graded polynomial identities as identities of universal algebras. Linear Algebra Appl. 562 (2019), 1–14.
  6. A. Bianchi, D. Diniz. Identities and isomorphisms of finite-dimensional graded simple algebras. J. Algebra 526 (2019), 333–344.
  7. O. Di Vincenzo, P. Koshlukov, A. Valenti. Gradings on the algebra of upper triangular matrices and their graded identities. J. Algebra 275, 2 (2004), 550–566.
  8. V. Drensky, E. Formanek. Polynomial Identity Rings. Adv. Courses Math. CRM Barcelona. Basel, Birkhäuser Verlag, 2004.
  9. R. Freese, R. McKenzie. Residually small varieties with modular congruence lattices. Trans. Amer. Math. Soc. 264, 2 (1981), 419–430.
  10. D. J. Gonçalves, E. Riva. Graded polynomial identities for the upper triangular matrix algebra over a finite field. J. Algebra 559 (2020), 625–645.
  11. D. Hobby, R. McKenzie. The structure of finite algebras. Contemp. Math., vol. 76, Providence, RI, American Mathematical Society, 1988.
  12. B. Jónsson. Algebras whose congruence lattices are distributive. Math. Scand. 21 (1967), 110–121.
  13. P. Koshlukov, M. Zaicev. Identities and isomorphisms of graded simple algebras. Linear Algebra Appl. 432, 12 (2010), 3141–3148.
  14. L. Kovács, M. Newman. Minimal verbal subgroups.
  15. R. L. Kruse. Identities satisfied by a finite ring. J. Algebra 26 (1973), 298–318.
  16. A. Kushkulei, Yu. Razmyslov. Varieties generated by irreducible representations of Lie algebras. Vestnik Moskov. Univ. Ser. I Mat. Mekh. 5, 1983, 4–7 (in Russian); English translation in: Moscow Univ. Math. Bull. 38, 5 (1983), 56–63.
  17. I. L’vov. Varieties of associative rings. I. Algebra i Logika 12 (1973), 269–297, 363 (in Russian); English translation in: Algebra and Logic 12, 3 (1973), 150–167 (1974).
  18. I. V. L’vov. Varieties generated by finite alternative rings. Algebra i Logika 17, 3 (1978), 282–286, 358 (in Russian); English translation in: Algebra and Logic 17, 3 (1978), 195–198.
  19. I. V. L’vov. Finite-dimensional algebras with infinite identity bases. Sibirsk. Mat. Ž. 19, 1 (1978), 91–99, 237 (in Russian); English translation in: Siberian Math. J. 19, 1 (1978), 63–69.
  20. Ju. A. Medvedev. Identities of finite Jordan Φ-algebras. Algebra i Logika 18, 6 (1979), 723–748, 755 (in Russian); English translation in: Algebra and Logic 18, 6 (1979), 460–477 (1980).
  21. Ju. A. Medvedev. Cross varieties of algebras. Mat. Sb. (N.S.) 115(157), 3 (1981), 391–425, 495496 (in Russian); English translation in: Math. USSR-Sb. 43, 3 (1982), 347–376.
  22. E. Neher. Polynomial identities and nonidentities of split Jordan pairs. J. Algebra 211, 1 (1999), 206–224.
  23. H. Neumann. Varieties of groups. Ergebnisse der Mathematik und ihrer Grenzgebiete. 2. Folge, Band 37. (Softcover reprint of the original 1st ed. 1967), Springer, 2012.
  24. S. Oates, M. B. Powell. Identical relations in finite groups. J. Algebra 1 (1964), 11–39.
  25. S. V. Polin. Identities of finite algebras. Sibirsk. Mat. Ž. 17, 6 (1976), 1356–1366, 1439 (in Russian); English translation in: Siberian Math. J. 17, 6 (1976), 992–999 (1977).
  26. Yu. P. Razmyslov. Identities of algebras and their representations. Transl. Math. Monogr., vol. 138. Providence, RI, American Mathematical Society, 1994.
  27. I. Shestakov, M. Zaicev. Polynomial identities of finite dimensional simple algebras. Comm. Algebra 39, 3 (2011), 929–932.