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

Irwing Kaplansky

A criterion of elementary divisor domain for distributive domains

Vasylyna Bokhonko

Department of Algebra and Logic

Faculty of Mechanics and Mathematics
Ivan Franko National University of L'viv

Abstract:

Under a ring  we understand an associative ring with 1.

The matrices A and B over a ring R are equivalent if exist invertible matrices P and Q of appropriate sizes such that  B=PAQ.

A ring R is called clean if every element of R is the sum of an idempotent and a unit.

A ring R is called an exchange ring if for every element a∈R there exists an idempotent e∈R such that e∈aR, 1-e∈(1-aR).

A ring R is called right (left) distributive if every lattice right (left) ideal of ring R is distributive.

We say that a duo-ring R has neat range 1 if for every  a,b∈R such that aR+bR=R there exist an element t∈R such that a factor-ring R/(a+br)R is a clean ring.

Theorem 1. Any distributive Bezout domain is an elementary divisor domain if and only if it is a duo-domain of neat range 1.

References

  Zabavsky B.V. Diagonal reduction of matrices over rings. Math. Stud., Monograph Series, v. XVI, Lviv, 2012, 251 p.

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