On rings of integer-valued rational functions

2024/10/21 by Chems-Eddin, Mohamed Mahmoud, Feryouch, Badr, Mouanis, Hakima +1
#Commutative Algebra (math.AC) #FOS: Mathematics

paper · doi:10.48550/arxiv.2410.16142

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^R_B(E,D), which naturally extends the concepts of integer-valued polynomials, defined as Int^R_B(E,D) =\lbrace f ∈ B(X); f(E)⊆ D\rbrace. The notion of Int^R_B(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.

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