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), $a, b \in D$ be nonzero elements with $a \nmid b$, $(a, b)D = dD$ for some $d \in D$, $D_a = D/aD$ 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) $\frac{a}{d}$ is a prime element and $a \nmid d^{2}$, (ii) every nonzero ideal of $bD_a$ is a cancellation ideal, and (iii) $bD_a$ is a field.