Geodesic
On Bhargava rings
Mathematica Bohemica, Tome 148 (2023) no. 2, pp. 181-195
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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 $\mathbb {B}_x(D):=\{f\in \nobreak K[X]\colon \text {for all}\ a\in D,\ f(xX+a)\in D[X]\}$. 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.
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 $\mathbb {B}_x(D):=\{f\in \nobreak K[X]\colon \text {for all}\ a\in D,\ f(xX+a)\in D[X]\}$. 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.
DOI : 10.21136/MB.2022.0137-21
Classification : 13B30, 13C11, 13C15, 13F05, 13F20
Keywords: Bhargava ring; localization; (locally) essential domain; locally free module; (faithfully) flat module; Krull dimension 
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Chems-Eddin, Mohamed Mahmoud; Ouzzaouit, Omar; Tamoussit, Ali. On Bhargava rings. Mathematica Bohemica, Tome 148 (2023) no. 2, pp. 181-195. doi: 10.21136/MB.2022.0137-21

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