ON A THEOREM OF ANDERSON AND CHUN

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 E 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 E 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.