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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

On common properties of Bezout domains and rings of matrices over them II

Volodymyr Shchedryk

Department of Algebra
Pidstryhach Institute for Applied Problems
of Mechanics and Mathematics

National Academy of Sciences of Ukraine

 

Abstract:

Let R be an commutative Bezout domain and D∈ Mn(R).  Denote Annr(D) by the set of right annihilator of the matrix D:

Annr(D)={ Q∈ Mn(R) | DQ=0 }

Denote (A,B)l by and [A,B] the left g.c.d. and right l.c.m. of matrices A, B respectively.

Theorem 1. Let A,B∈ Mn(R). There exists invertible matrix such that 

where D=(A,B)l  and

Annr(D)⊆ Annr(V).

Theorem 2. The matrix is associated by right to matrix

Corollary 1. The multiple of not zero diagonal elements of right Hermite normal form of matrices A, B is coincide with multiple of not zero diagonal elements of right Hermite normal form of (A,B)l , [A,B]r.

Corollary 2.

det(AB)=det(A,B)l det[A,B]r.

 

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