On a class of half-factorial domains

Let $R$ be an integral domain. For elements $a,b\in R$, let $[a,b]$ denote their greatest common divisor, if it exists. We say that $R$ has the Z-property if whenever $a,b,c,d$ and $e$ are nonzero nonunits of $R$ such that $abc=de$, then $[ab,d]\mathrm{\ne }1$ or $[ab,e]\mathrm{\ne }1$. The purpose of this paper is to study this property. The atomic integral domains that have this property constitute a class of half-factorial domains. Also, it is known that $R$ must have this property in order for the polynomial ring $R[x]$ to be half-factorial. We use it to give a characterization of half-factorial polynomial rings in the case where every $v$-ideal is $v$-generated by two elements. We also show that if $R$ is a Krull domain with this property, then $R$ has torsion class group.