ON A THEOREM OF ANDERSON AND CHUN

Abstract

A commutative ring R is called strongly regular associate if, for any a,b∈R, Ra=Rb implies that a=rb and sa=b for some regular elements r,s∈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∈R, Ra+Rb=R implies that a+bx is a regular for some x∈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.