Non-commutative Bézout domains and pseudo MV-algebras

Motivated by the successful applications of Bézout domains in the theory of MV-algebras, we develop a generalization of Bézout domains to the non-commutative case for applications to pseudo MV-algebras. The well-known Kaplansky-Jaffard-Ohm theorem is generalized by constructing (not necessarily commutative) domains of Bézout type whose groups of divisibility are certain lattice-ordered (non-Abelian) groups. Some related ring properties (like Ore conditions) are also studied, and their connections to pseudo MV-algebras are established. A few applications are given to illustrate how our results can be applied to certain pseudo MV-algebras while they are treated as subsets of unital ℓ-groups.