On Bhargava rings

. Let D be an integral domain with the quotient field K , X an indeterminate over K and x an element of D . The Bhargava ring over D at x is defined to be B x ( D ) := { f ∈ K [ X ]: for all a ∈ D, f ( xX + a ) ∈ D [ X ] } . In fact, B x ( D ) is a subring of the ring of integer-valued polynomials over D . In this paper, we aim to investigate the behavior of B x ( D ) under localization. In particular, we prove that B x ( D ) behaves well under localization at prime ideals of D , when D is a locally finite intersection of localizations. We also attempt a classification of integral domains D such that B x ( D ) is locally free, or at least faithfully flat (or flat) as a D -module (or D [ X ]-module, respectively). Particularly, we are interested in domains that are (locally) essential. A particular attention is devoted to provide conditions under which B x ( D ) is trivial when dealing with essential domains. Finally, we calculate the Krull dimension of Bhargava rings over MZ-Jaffard domains. Interesting results are established with illustrating examples.