Algebras of quotients of Jordan algebras
The author gives an analogue of Lambek and Utumi's associative algebra of quotients for Jordan algebras over arbitrary commutative rings of scalars, proving the existence (and uniqueness up to isomorphism extending the identity on \(J\)) of the maximal algebra of quotients for nondegenerate Jordan algebras. Algebras of quotients are defined in terms of inner ideals of denominators satisfying the property of being dense, a notion also introduced in this work which is the Jordan analogue to dense one-sided ideals appearing in the associative localization theory. Following a structural approach, the author first proves that the maximal algebra of quotients of a nondegenerate Jordan PI-algebra is the so-called almost classical algebra of quotients. Such a construction consists of an extension to quadratic Jordan algebras of the nearly classical localization of an algebra [see \textit{K. I. Beidar} and \textit{A. V. Mikhalev}, J. Sov. Math. 47, No. 3, 2525--2536 (1989); translation from Tr. Semin. Im. I. G. Petrovskogo 12, 59--74 (1987; Zbl 0683.17016)] also called almost classical localization by \textit{R. Wisbauer} [Modules and algebras: bimodule structure and group actions on algebras. Pitman Monographs and Surveys in Pure and Applied Mathematics. 81. Harlow, Essex: Addison Wesley Longman (1996; Zbl 0861.16001)], and the proof is based on the good relationship between the weak center of a Jordan algebra and that of its algebras of quotients. On the other hand the proof of the existence of maximal algebras of quotients for nondegenerate Jordan non-PI-algebras (and therefore special) is based on the relationship between the dense inner ideals of a special Jordan algebra and the dense one-sided ideals of its \(\ast\)-tight associative envelopes. In this case the maximal algebra of quotients turns out to be an ample subspace of symmetric elements of the maximal algebra of symmetric quotients of a \(\ast\)-tight associative envelope of the Jordan algebra. Both results are then put together to prove the existence of the maximal algebras of quotients of nondegenerate Jordan algebras. The author also provides a thorough overview of other different notions of Jordan quotients, which have previously appeared in the literature, showing how they fit into the maximal algebra of quotients.
scientific article; zbMATH DE number 5718175
scientific article