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
Let R be a commutative Bezout domain in which for any three relatively prime nonzero elements a, b, c there exist such element r, that (a+br,c)=1. And let A1, A2, A3 be relatively prime on left side nonzero matrices from M2(R). Then there exist such matrix T and i,j,k ∈ , that Ai+AjT, Ak are relatively prime on left side
Department of Algebra and Logic
Faculty of Mechanics and Mathematics
Ivan Franko National University of L'viv
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