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 understood as a associative ring with nonzero unit element.
A ring R is said to have stable range 1, if for any a, b є R satisfying aR + bR = R, there exists t є R such that a + bt is an invertible element in R.
A ring R is called a exchange ring if for any element a є R there exists an idempotent e є R such that e є aR and 1-e є (1-a)R.
We will denote the Jacobson radical of a ring R by J(R). A ring R is said to be a semiexchange ring if the factor ring R/J(R) is an exchange ring.
Let R be a ring and a, b є R. The pair (a,b) is said to be an e-atomic pair if there exist Q є GE2(R) and an atom element q є R such that (a,b)Q=(q,m) for a m є R. Then R is said to be e-atomic if for any a,b є R such that aR+bR=R and 0≠ c є R, there exists y є R such that (a+by,c) is an e-atomic pair.
A matrix ring of order n over a ring R is denoted by Mn(R). We say that a matrix A є Mn(R) is full if Mn(R) A Mn(R) = Mn(R). We denote by F(Mn(R)) the class of all full matrices of the ring Mn(R).
Theorem 1. A Bezout duo ring with stable range 1 is a ring with elementary reduction of matrices.
Theorem 2. Let R be a semiexchange quasi-duo Bezout ring. Then R is a ring with elementary reduction of matrices if and only if it is a duo.
Theorem 3. Let R be a commutative e-atomic ring. The following statements are equivalent:
(1) R - ring with elementary reduction of matrices;
(2) R - Bezout ring.
Theorem 4. Let R be a commutative elementary divisor ring and n є Z>1. Then for any full matrices A,B є F(Mn(R)), B≠O, there exists a right (left) 2(n-1)-stage terminating division chain in Mn(R).
Theorem 5. Let R is a PID and n є N>1. Then Mn(R) is right (left) 2-stage euclidean set.