This paper investigates three interconnected problems in the theory of factorizations and multiplicative properties: analogues of Jacobian-type conditions in noncommutative settings, the existence problem of the so-called middle divisors in square-free and radical factorizations (both in commutative and noncommutative frameworks), and square-free ideals in noncommutative rings. For submonoids M of l-GCD monoids M endowed with local normalization, we obtain equivalent inclusion forms \({{\,\textrm{Sqf}\,}}_L M \subseteq {{\,\textrm{Sqf}\,}}_L H\) and \({{\,\textrm{Irr}\,}}_L M \subseteq {{\,\textrm{Sqf}\,}}_L H\) , which serve as noncommutative counterparts of Jacobian-type conditions. We then develop and compare criteria for the existence of middle divisors, showing their connections with earlier 4s–6s criteria in both commutative and noncommutative versions. In the final part, we present extensions of the theory of square-free ideals to noncommutative rings, including square-testing criteria and examples that separate the notions of square-free and radical ideals. The results identify minimal assumptions required to preserve factorial-style inclusions and provide tools for further exploration of algebraic aspects of the Jacobian conjecture in noncommutative contexts.
