부산대학교 도서관

The authors introduce and study the following notions and their equivalents. They call $R$ a GD(1) (respectively, GD(2)) domain if every nonzero element of $R$ is contained in only a finite number of principal prime (respectively, prime) ideals. They characterize a GD(1) domain as a domain $R$ such that for each ideal $I$ of the polynomial ring $R[X]$ with $I\cap R=(0)$ only finitely many prime elements of $R$ become units in $R[X]/I.$ They characterize a GD(2) domain as a domain $R$ such that for each ideal $J$ of $R[X]$ and for each $I\subseteq J$ with $I\cap R=(0)$ there are only a finite number of prime ideals of $R$ that satisfy the equation $PR[X]+I=J$. They show that $R$ is GD(1) if and only if the polynomial ring $R[X]$ is GD(1) and that $R[X]$ is GD(2) if and only if $R$ is a field. Selective examples of ${\rm GD}(i)$ and non-${\rm GD}(i)$ domains are offered. The paper is interesting reading from the techniques point-of-view and for its wealth of references.