On Bhargava rings

Abstract

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 $\mathbb{B}_x(D):=\{f\in\nobreak K[X] \text{for all} a\in D, f(xX+a)\in D[X]\}$. In fact, $\mathbb{B}_x(D)$ is a subring of the ring of integer-valued polynomials over $D$. In this paper, we aim to investigate the behavior of $\mathbb{B}_x(D)$ under localization. In particular, we prove that $\mathbb{B}_x(D)$ 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 $\mathbb{B}_x(D)$ 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 $\mathbb{B}_x(D)$ 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.

Keywords

• bhargava ring
• localization
• (locally) essential domain
• locally free module
• (faithfully) flat module
• krull dimension