Let A,B∈ Mn(R). There exists invertible matrix such that where D=(A,B)l and Annr(D)⊆ Annr(V).
Theorem 1. Let R be a commutative Bezout ring a stable range 2. Let A, B∈ F(R2) be such that AR2+BR2=R2, moreover, the matrix B admits a diagonal reduction. Then there exists a full matrix T∈ F(R2) such that A+BT is an invertible matrix. Theorem 2. Let R be a commutative elementary divisor ring. If A, B∈ F(R2) and AR2+BR2=R2, then there exists full matrix T∈ F(R2) such that A+BT is an invertible matrix.
It will be shown that the commutative Bezout domain is an elementary divisor ring if and only if its stable rang equal to half. If commutative Bezout domain is an elementary divisor ring rhen some localization of matrix ring over it is an exchange ring
Will present and discuss interesting open problems from various branches of algebra and its applications.
Department of Algebra and Logic
Faculty of Mechanics and Mathematics
Ivan Franko National University of L'viv
1 Universytetska Str., 79000 Lviv, Ukraine
Tel: (+380 322) 394 172
E-mail: oromaniv at franko.lviv.ua