LAUSR

A BSTRACT . 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.