Two notes on homogeneous prime ideals in graded Noetherian rings

The main results in this paper give: (1) a characterization for when there exist infinitely many homogeneous prime ideals p between two given homogeneous prime ideals P Q such that ht(Q/P)=2 in a -graded Noetherian ring; and (2) a characterization of when a given homogeneous prime ideal Q in an arbitrary Noetherian Rees ring contains infinitely many homogeneous prime ideals p such that ht(p)=ht(Q)−1.