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).

For any n ≥ 2 and fixed k ≥ 1, we give necessary and sufficient conditions for an arbitrary nonzero square matrix in the matrix ring Mn(F) to be written as a sum of an invertible matrix U and a nilpotent matrix N with N k = 0 over an arbitrary field F.

We define the socle of a nondegenerate Lie algebra as the sum of all its minimal inner ideals. The socle turns out to be an ideal which is a direct sum of simple ideals, and satisfies the descending chain condition on principal inner ideals. Every classical finite dimensional Lie algebra coincides with its socle, while relevant examples of infinite dimensional Lie algebras with nonzero socle are the simple finitary Lie algebras and the classical Banach Lie algebras of compact operators on an infinite dimensional Hilbert space. This notion of socle for Lie algebras is compatible with the previous ones for associative algebras and Jordan systems. We conclude with a structure theorem for simple nondegenerate Lie algebras containing abelian minimal inner ideals, and as a consequence we obtain that a simple Lie algebra over an algebraically closed field of characteristic 0 is finitary if and only if it is nondegenerate and contains a rank-one element.

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