When are Associates Unit Multiples?

Abstract

Let R be a commutative ring with identity. For a, b R define a and b to be associates, denoted a b, if a|b and b|a, to be strong associates, denoted a b, if a = ub for some unit u of R, and to be very strong associates, denoted by a = b, if a b and further when a = 0, a = rb implies that r is a unit. Certainly a = b a b a b. In this paper we study commutative rings R, called strongly associate rings, with the property that for a, b R, a b implies a b and commutative rings R, called prsimplifiable rings, with the property that for a, b R, a b (or a b) implies that a = b.