ON A THEOREM OF ANDERSON AND CHUN

Abstract

A commutative ring $R$ is called strongly regular associate if, for any $a,b\in R$, $Ra=Rb$ implies that $a=rb$ and $sa=b$ for some regular elements $r,s\in R$. In this paper, we first give a characterization of strongly regular associate rings. A ring $R$ is said to have regular range 1 if, for any $a,b\in R$, $Ra+Rb=R$ implies that $a+bx$ is a regular for some $x\in R$. We show that the ring of continuous functions $C(X)$ is strongly regular associate if and only if it has regular range 1. Finally, we generalize a theorem of Anderson and Chun, which states that $C([a,b])$ is a strongly regular associate ring.