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 $\mathbb{B}_x(D):=\{f\in\nobreak K[X] \text{for all} a\in D, f(xX+a)\in D[X]\}$. In fact, $\mathbb{B}_x(D)$ is a subring of the ring...
| Main Authors: | , , |
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| Format: | Article |
| Language: | English |
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Institute of Mathematics of the Czech Academy of Science
2023-07-01
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| Series: | Mathematica Bohemica |
| Subjects: | |
| Online Access: | http://mb.math.cas.cz/full/148/2/mb148_2_3.pdf |
| Summary: | 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. |
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| ISSN: | 0862-7959 2464-7136 |