# A Dirichlet ring as a ring of neat range 1. Ch.I

- Written by B. Zabavsky
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Professor

Department of Algebra and Logic

Faculty of Mechanics and Mathematics

Ivan Franko National University of L'viv

Abstract:

**Definition**. *A commutative domain R is said to be a Dirichlet ring if for any relative prime elements a,b*

**∈**

**R**there exists**t****∈**

**R**such that**a+bt**is an atom of a ring**R**Examples of a Dirichlet ring is ** Z**,

**,**

*R=*{*z*∈_{0}+a_{1}+....+a_{n}+... | z_{0}*Z, a*∈_{i}*Q*}**where**

*P[x]***is a finite field.**

*P***not is a Dirichlet ring.**

*C[x]***Definition**. * A commutative ring R is said to be of neat range 1 if for any relatively prime elements a,b*

**∈**

**R**there exists**t****∈**

**R**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?*

* *