Jordan systems of Martindale-like quotients

In this paper we introduce the notion of Jordan system (algebra, pair or triple system) of Martindale-like quotients with respect to a filter of ideals as that whose elements are absorbed into the original system by ideals of the filter, and prove that it inherits regularity conditions such as (semi)primeness and nondegeneracy. When we consider power filters 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.