On rings of integer-valued rational functions

Abstract

Let D ⊆ B be an extension of integral domains and E a subset of the quotient field of D. We introduce the ring of D -valued B-rational functions on E, denoted by Int B R ( E , D ) , which naturally extends the concept of integer-valued polynomials, defined as Int B R ( E , D ) : = { f ∈ B ( X ) ; f ( E ) ⊆ D } . The notion of Int B R ( E , D ) boils down to the usual notion of integer-valued rational functions when the subset E is infinite. In this paper, we aim to investigate various properties of these rings, such as prime ideals, localization, and the module structure. Furthermore, we study the transfer of some ring-theoretic properties from Int R ( E , D ) to D.