Cleanness of a Dubrovin valuation ring

Abstract

An order R in a simple Artinian ring Q is said to be a Dubrovin valuation ring if R is Bezout and R/J(R) is a simple Artinian, where J(R) is the Jacobson radical of R. A ring R with unity is called clean, if every element x ∈ R is clean i.e. for every element x ∈ R there exist an idempotent element e ∈ R and a unit element u ∈ R such that x=e+u. In this article, it will be investigated some properties of clean Dubrovin valuation ring and give some examples related to a Dubrovin valuation ring and a clean ring.An order R in a simple Artinian ring Q is said to be a Dubrovin valuation ring if R is Bezout and R/J(R) is a simple Artinian, where J(R) is the Jacobson radical of R. A ring R with unity is called clean, if every element x ∈ R is clean i.e. for every element x ∈ R there exist an idempotent element e ∈ R and a unit element u ∈ R such that x=e+u. In this article, it will be investigated some properties of clean Dubrovin valuation ring and give some examples related to a Dubrovin valuation ring and a clean ring.