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We prove that the class of full matrices of order 2 over elementary divisor ring has the stable range 1. [ABSTRACT FROM AUTHOR]

We prove that a sharp Bézout domain is an elementary divisor ring. [ABSTRACT FROM AUTHOR]

We prove that, in a domain of elementary divisors, the intersection of all nontrivial two-sided ideals is equal to zero. We also show that a Bézout domain with finitely many two-sided ideals is a domain of elementary divisors if and only if it is a 2-simple Bézout domain. [ABSTRACT FROM AUTHOR]

It is known that a simple Bézout domain is the domain of elementary divisors if and only if it is 2-simple. The block-diagonal reduction of matrices over an n -simple Bézout domain ( n ≥ 3) is realized. [ABSTRACT FROM AUTHOR]

We introduce the notion of a relatively adequate element of a commutative ring, which is not necessary a domain, investigate properties of such elements, and on the basis of this, propose the characterization of absolutely adequate elements. In particular, we prove that an adequate Bezout ring has a stable rank that is not higher than 2. As a consequence, it is stated that the adequate Bezout ring is a ring of elementary divisors. [ABSTRACT FROM AUTHOR]

It is shown that an adequate ring with nonzero Jacobson radical has a stable range 1. A class of matrices over an adequate ring with stable range 1 is indicated. [ABSTRACT FROM AUTHOR]