Beitr Algebra Geom https://doi.org/10.1007/s13366-020-00514-7 ORIGINAL PAPER Note on integer-valued polynomials on a residually cofinite subset Ali Tamoussit 1 Received: 4 April 2020 / Accepted: 24 June 2020 © The Managing Editors 2020 Abstract Let D be an integral domain with quotient field K , E a subset of K and X an indeterminate over K . The ring Int ( E , D) := { f ∈ K [ X ] : f ( E ) ⊆ D}, of integer-valued polynomials on E with respect to D, is known to be a D-algebra. Obvi- ously, Int ( D, D) = Int ( D), is the classical ring of integer-valued polynomials over D. In this paper, we study the faithful flatness, the local freeness and the calculation of the Krull dimension of integer-valued polynomials on a residually cofinite subset over locally essential domains. In particular, we prove that Int ( E , D) is faithfully flat as a D-module and is of Krull dimension less than that of D[ X ], when D is a locally essential domain and E ⊆ D is residually cofinite with D. Also, we get stronger results in the case of domains that are either almost Krull or t -almost Dedekind. Keywords Integer-valued polynomials · Faithfully flat modules · Locally free modules · Locally essential domains · Krull dimension Mathematics Subject Classification 3F20 · 13C11 · 13C15 · 13F05 1 Introduction Throughout this paper, D will denote an integral domain D with quotient field K . The polynomials with coefficient in K that take values in D form a commutative D-algebra, denoted by Int ( D), that is, Int ( D) := { f ∈ K [ X ]: f ( D) ⊆ D}, and called the ring of integer-valued polynomials over D. Cahen (1993) considered a generalization of Int ( D), namely, the ring of integer- valued polynomials over a subset E of K , to be Int ( E , D) := { f ∈ K [ X ]: f ( E ) ⊆ D}. Obviously, Int ( D, D) = Int ( D) and D ⊆ Int ( E , D) ⊆ K [ X ]. B Ali Tamoussit tamoussit2009@gmail.com 1 Department of Mathematics, The Regional Center for Education and Training Souss Massa, P.O. Box 106, Inezgane, Morocco 123