Elementary divisor rings with Dubrovin-Komarnytsky property^*

Elementary divisor rings were first introduced by Kaplansky in his seminal work. The purpose of this research is to extend Kaplansky’s study of commutative elementary divisor rings to certain classes of associative rings under weaker conditions than commutativity. We introduce two new classes of non-commutative rings: those with the DK-property (Dubrovin–Komarnytsky property) and those with the D-property (Dubrovin property), and investigate the structure of elementary divisor rings within these settings. Our main focus is on non-commutative rings of stable range 1. For such rings, we develop a theory of reduction matrices, which allows us to construct and analyze new families of non-commutative elementary divisor rings. In addition, we introduce the concept of an elementary element in a non-commutative ring. We prove that for Bézout domains of stable range 1 with the DK-property, a ring R is an elementary divisor ring if and only if every nonzero element of R is elementary.