From Jacobian-Type Conditions to Noncommutative Square-Free and Radical Factorizations

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 \,Sqf\,L M ⊆ \,Sqf\,L H Sqf L M ⊆ Sqf L H and \,Irr\,L M ⊆ \,Sqf\,L H Irr L M ⊆ 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 4 s –6 s 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.