Rings of the right (left) almost stable range $1$

Abstract

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 $a\in R$ 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 $1$.