Rings of the right (left) almost stable range 1

We introduce a concept of rings of right (left) almost stable range 1 and we construct a theory of a canonical diagonal reduction of matrices over such rings. A description of new classes of noncommutative elementary divisor rings is done as well. In particular, for Bézout D-domain we introduced the notions of D-adequate element and D-adequate ring. We proved that every D-adequate Bézout domain has almost stable range 1. For Hermite D-ring we proved the necessary and sufficient conditions to be an elementary divisor ring. A ring R is called an L-ring if the condition RaR = R for some a ∈ R implies that a is a unit of R. We proved that every L-ring of almost stable range 1 is a ring of right almost stable range 1.