Kaplansky-type Theorems, II

Let D be an integral domain with quotient field K, X be an indeterminate over D, and D[X] be the polynomial ring over D. A prime ideal Q of D[X] is called an upper to zero in D[X] if Q = fK[X] D[X] for some f D[X]. In this paper, we study integral domains D such that every upper to zero in D[X] contains a prime element (resp., a primary element, a t-invertible primary ideal, an invertible primary ideal).