Jordan systems of Martindale-like quotients

Abstract

In this paper we introduce the notion of Jordan system (algebra, pair or triple system) of Martindale-like quotients with respect to a ÿlter of ideals as that whose elements are absorbed into the original system by ideals of the ÿlter, and prove that it inherits regularity conditions such as (semi)primeness and nondegeneracy. When we consider power ÿlters of sturdy ideals, the notions of Jordan systems of Martindale-like quotients and Lie algebras of quotients are related through the Tits-Kantor-Koecher construction, and that allows us to give constructions of the maximal systems of quotients when the original systems are nondegenerate.The theory of rings of quotients has its origins between 1930 and 1940, in the works of Ore and Osano on the construction of the total ring of fractions. In that decade Ore proved that a necessary and su cient condition for a ring R to have a (left) classic ring of quotients is that for any regular element a in R, and any b ∈ R there exist a regular c ∈ R and d ∈ R such that cb = da (left Ore condition).

Maximal algebras of Martindale-like quotients of strongly prime linear Jordan algebras

“…In [6,14] algebras of quotients in the Jordan and associative cases are introduced and studied. When dealing with associative algebras, algebra of quotients will mean Martindale algebra of symmetric quotients [14].…”

“…When dealing with associative algebras, algebra of quotients will mean Martindale algebra of symmetric quotients [14]. When dealing with Jordan algebras, algebra of quotients will mean Jordan algebra of Martindale-like quotients [6].…”

“…When J is strongly prime, the above filter is just the set of all nonzero ideals of J . [6,14], we will outline the monomorphism linking an algebra with its algebra of quotients. Thus, given an algebra J , an algebra of quotients for J with respect to a filter of ideals F is (Q, τ ), such that Q is an algebra, τ : J → Q is an algebra monomorphism, and, for any 0 = q ∈ Q, there exists I ∈ F such that…”

“…In [6], definitions of linear Jordan systems of Martindale-like quotients are given. When 1/3 and 1/2 are assumed to be in the ring of scalars, the existence of maximal systems of quotients is obtained by using Lie algebras of quotients [20] through the Tits-KantorKoecher construction.…”

“…When 1/3 and 1/2 are assumed to be in the ring of scalars, the existence of maximal systems of quotients is obtained by using Lie algebras of quotients [20] through the Tits-KantorKoecher construction. The notions of Jordan system of quotients in [6] follow the pattern of McCrimmon [14], and are given with respect to filters of ideals, so that some general properties are obtained when the filters are sufficiently regular.…”