Department of Algebra and Logic
Faculty of Mechanics and Mathematics
Ivan Franko National University of L'viv
email: andrij.sagan at gmail.com
A ring R is a associative ring with nonzero identity. A right (left) Bezout ring is a ring in which every nitely generated right (left) ideal is principal, and if both R is a Bezout ring. A ring R is said to be right (left) semihereditary, if any right (left) nitely generated ideal in R is projective. An elementary n×n matrix with entries from R is a square n×n matrix of one of the types below: 1) diagonal matrix with invertible diagonal entries; 2) identity matrix with one additional non diagonal nonzero entry; 3) permutation matrix, i.e. result of switching some columns or rows in the identity matrix.
If an arbitrary matrix over R admits elementary reduction, then R is called a ring with elementary reduction of matrices. A ring R is a stable range 1 ring, if for every elements a, b ∈ R such that aR + bR = dR, there is an element t ∈ R such that a + bt is an invertible element in R.
Theorem 1. Every right semihereditary Bezout duo-ring with stable range 1 is a ring with elementary reduction of matrices.
Theorem 2. Let R be a right semihereditary duo-ring. Then the following are equivalent:
1) R is a quasi-euclidean ring;
2) R is a Bezout ring and GLn(R) = GEn(R) for every positive integer n≥2.