Abstract
It is shown that the center of a nondegenerate, purely alter native algebra A contains a dense ideal I such that for any nonzero element t ∈I the classical localization A t of the algebra A with respect to the element t is a Cayley—Dickson algebra over its center. It is established that the classical ring of quotients of an alternative PI-algebra is a PI-algebra. The obtained results are applied to the description of von Noumann regular alternative algebras.
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Literature Cited
K. I. Beidar, "Rings with generalized identities. I," Vestn. Moskov. Univ., Ser. I, Mat. Mekh., No. 2, 19–26 (1977).
K. I. Beidar, "Rings with generalized identities. III," Vestn. Mosk. Univ., Ser. I, Mat. Mekh., No. 4, 66–73 (1978).
K. I. Beidar, "Classical rings of quotients of PI-algebras," Usp. Mat. Nauk,33, No. 6, 197–198 (1978).
K. I. Beidar and A. V. Mikhalev, "Orthogonal completeness and algebraic systems," Usp. Mat. Nauk,40, No. 6, 79–115 (1985).
K. I. Beidar and A. V. Mikhalev, "The structure of nondegenerate alternative algebras," Trudy Sem. Petrovsk. No. 12, 59–74 (1987).
K. A. Zhevlakov, "On radicals and von Neumann ideals," Algebra Logika,8, No. 3, 425–439 (1969).
K. A. Zhevlakov, A. M. Slin'ko, I. P. Shestakov, and A. I. Shirshov, Rings That Are Nearly Associative, Academic Press, New York (1982).
I. N. Herstein, Noncommutative Rings, The Carus Mathematical Monographs, No. 15, The Math. Assoc. America (1968).
S. A. Amitsur, "Rings with involution," Israel J. Math.,6, No. 2, 99–106 (1968).
L. H. Rowen, "On rings with central polynomials," J. Algebra,31, No. 3, 393–426 (1974).
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Translated from Trudy Seminara imeni I. G. Petrovskogo, No. 16, pp. 227–235, 1992.
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Beidar, K.I. Classical localizations of alternative algebras. J Math Sci 69, 1098–1104 (1994). https://doi.org/10.1007/BF01254395
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DOI: https://doi.org/10.1007/BF01254395