Department of Algebra and Logic
Faculty of Mechanics and Mathematics
Ivan Franko National University of L'viv
Under a ring we introduce a commutative ring R with 1≠0. A Bezout ring is the ring in which every finitely generated ideal is principal.
Definition 1. A commutative Bezout ring R we call a Toeplitz ring if for any a,b ϵ R there exists T − an invertible Toeplitz matrix such that (a,b)T=(d,0).
Теорема 1. Let R is a commutative Hermite ring of stable range one. Then R is the Toeplitz ring.
Definition 2. A ring R called ring of unit square stable range one if from condition aR+bR=R is following that there exists a unit element t such that a2+bt − the unit element of the ring R.
Theorem 2. Let R is a commutative Hermite ring unit square stable range one. Then any matrix of the second order over R is reduced to diagonal form by the invertible Toeplitz matrices.
Theorem 3. Let R a commutative Bezout ring of stable range one. Then:
1) some unimodular row (column) of length 2 over R is reduced to the invertible Toeplitz matrix;
2) some matrix of the second order over R is reduced to canonical diagonal form by the right and left multiplication invertible Toeplitz matrices;
3) some invertible matrix of the second order over ring R is decomposed in product the invertible Toeplitz matrices.