Jordan-Hölder, modularity and distributivity in non-commutative algebra (Q861861)

The authors study weakly exact categories as a part of non-commutative homological algebra. The main concern are these weakly exact categories for which the lattice of subobjects of an object in the categories has a structure of \(w\)-modular \(w\)-lattice. Thus the subobjects and subquotients of objects in such a category are investigated in details. The main result says that the free \(w\)-modular \(w\)-lattice generated by two (finite) chains with normality conditions is weakly distributive. As a consequence, the authors show that the Jordan-Hölder theorem for \(w\)-exact categories holds true. As examples of weakly exact categories such that the lattice of subobjects of an object is a weakly lattice, one may think of the category of groups and the category of rings (without unit assumption). Roughly speaking, in the paper a ``weakly'' version for subobjects in weakly exact categories is established. The paper is quite interesting.