Abstract
For an integral domain R with quotient field K, an upper-type ideal of R[x] is an ideal of the form \({I_f = f({\rm x})K[{\rm x}] \cap R[{\rm x}]}\) for some polynomial \({f({\rm x}) \in K[{\rm x}] \backslash K}\). Clearly, I f = I rf for each nonzero \({r \in R}\). Hence one can always choose f (x) from R[x]. Such an ideal I f is said to be almost principal if there is a nonzero element \({s \in R}\) such that \({sI_f \subseteq f({\rm x})R[{\rm x}]}\). If each upper-type ideal of R[x] is almost principal, then R[x] is said to be an almost principal ideal domain. The primary objective of this paper is to provide a unifying technique for verifying that certain types of domains always have corresponding polynomial rings that are almost principal ideal domains. These same techniques also will be used to show that certain types of upper-type ideals are always almost principal.
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