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"I liked the algebraic way of looking at things.
I’m additionally fascinated when the algebraic method is applied to infinite objects”.

Irwing Kaplansky

Reduction matrices over Bezout rings

 

 

Andrij Sagan

Department of Algebra and Logic
Faculty of Mechanics and Mathematics
Ivan Franko National University of L'viv
email: andrij.sagan at gmail.com

 
 

Abstract:

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.

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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

Scientific seminar, May, 2016