Abstract
Let Φbe an associative ring with 1, containing 1/6,and let A be an arbitrary Mal'tsev Φ-algebra. For some fully invariant ideals of an algebra A, we prove that their products lie in the annihilator Annof this algebra. As a consequence, it is inferred that for every algebra A over a field Φof characteristic 0, satisfying the nth Engel condition, there exists an N such that \(A^N \cdot A^2 \subseteq Ann^A \).Also, some identities in Mal'tsev algebras having faithful representations are proved to hold in separated Mal'tsev algebras; in the class of alternative algebras, new central and kernel functions are constructed.
Similar content being viewed by others
References
V. T. Filippov, “On Engelian Mal'tsev algebras,”Algebra Logika,15, No. 1, 89–109 (1976).
V. T. Filippov, “Prime Mal'tsev algebras,”Mat. Zametki,31, No. 5, 669–678 (1982).
V. T. Filippov, “Nilpotent ideals of Mal'tsev algebras,”Algebra Logika,18, No. 5, 599–613 (1979).
V. T. Filippov, “AT-ideal in Mal'tsev algebras,”Sib. Mat. Zh.,29, No. 3, 148–155 (1988).
V. T. Filippov, “Free Mal'tsev algebras and alternative algebras,”Algebra Logika,21, No. 1, 84–107 (1982).
E. T. Zel'manov, “Engelian Lie algebras,”Sib. Mat. Zh.,29, No. 5, 112–117 (1988).
V. T. Filippov, “Mal'tsev algebras satisfying Engel's condition,”Algebra Logika,14, No. 4, 441–455 (1975).
V. T. Filippov, “Varieties of Mal'tsev and alternative algebras generated by algebras of finite rank,” in:Groups and Other Algebraic Systems with Finiteness Conditions [in Russian], Nauka, Novosibirsk (1984), pp. 139–156.
V. T. Filippov, “Variety of Mal'tsev algebras,”Algebra Logika,20, No. 3, 300–314 (1981).
A. A. Sagle, “Malcev algebras,”Trans. Am. Math. Soc.,101, No. 3, 426–458 (1961).
V. T. Filippov, “The measure of non-Lieness for Mal'tsev algebras,”Algebra Logika,31, No. 2, 198–217 (1992).
Yu. A. Medvedev, “Example of a variety of solvable alternative algebras over a field of characteristic 2 having no finite bases of identities,”Algebra Logika,19, No. 3, 300–313 (1980).
V. T. Filippov, “Toward a theory of finitely generated Mal'tsev algebras,”Algebra Logika,19, No. 4, 480–499 (1980).
V. T. Filippov, “Semiprimary Mal'tsev algebras of characteristic 3,”Algebra Logika,14, No. 1, 100–111 (1975).
E. N. Kuz'min, “Mal'tsev algebras and their representations,”Algebra Logika,7, No. 4, 48–69 (1968).
E. N. Kuz'min, “Locally nilpotent radical of Mal'tsev algebras satisfying thenth Engel condition,”Dokl. Akad. Nauk SSSR,177, No. 3, 508–510 (1967).
Additional information
Translated fromAlgebra i Logika, Vol. 33, No. 5, pp. 576–595, September–October, 1994.
Rights and permissions
About this article
Cite this article
Filippov, V.T. Annihilators of Mal'tsev algebras. Algebr Logic 33, 322–334 (1994). https://doi.org/10.1007/BF00739573
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF00739573