"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 Dirichlet ring as a ring of neat range 1. Ch.I

Bogdan Zabavsky

Professor
Department of Algebra and Logic
Faculty of Mechanics and Mathematics
Ivan Franko National University of L'viv

Abstract:

DefinitionA commutative domain R is said to be a Dirichlet ring if for any relative prime elements a,bR there exists tR such that a+bt is an atom of a ring R

Examples of a Dirichlet ring is Z, R= {z0+a1+....+an+... | z0Z, aiQ}, P[x] where P  is a finite field. C[x] not is a Dirichlet ring.

Definition A commutative ring R is said to be of neat range 1 if for any relatively prime elements a,bR there exists tR such that for element a+bt=c a ring R/cR is a clean ring

Theоrem 1. A commutative Bezout ring R of stable range 2 is an elementary divisor ring iff R is a ring of neat range 1

Theоrem 2. A Dirichlet Bezout domain is an elementary divisor ring

Open question. Let R is a commutative Bezout domain in whict every maximal ideal is a principal. Under what conditions R is a Dirichlet ring?

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