When Does a Quotient Ring of a PID Have the Cancellation Property?

Abstract

. An ideal I of a commutative ring is called a cancellation ideal if IB = IC implies B = C for all ideals B and C . Let D be a principal ideal domain (PID), a,b ∈ D be nonzero elements with a (cid:45) b , ( a,b ) D = dD for some d ∈ D , D a = D/aD be the quotient ring of D modulo aD , and bD a = ( a,b ) D/aD ; so bD a is a nonzero commutative ring. In this paper, we show that the following three properties are equivalent: (i) ad is a prime element and a (cid:45) d 2 , (ii) every nonzero ideal of bD a is a cancellation ideal, and (iii) bD a is a field.