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 présimplifiable rings, with the property that for a, b ∈ R, a ∼ b (or a ≈ b) implies that a ≅ b. © 2004 Rocky Mountain Mathematics Consortium.