| Title:
|
On Bhargava rings (English) |
| Author:
|
Chems-Eddin, Mohamed Mahmoud |
| Author:
|
Ouzzaouit, Omar |
| Author:
|
Tamoussit, Ali |
| Language:
|
English |
| Journal:
|
Mathematica Bohemica |
| ISSN:
|
0862-7959 (print) |
| ISSN:
|
2464-7136 (online) |
| Volume:
|
148 |
| Issue:
|
2 |
| Year:
|
2023 |
| Pages:
|
181-195 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| 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 $\mathbb {B}_x(D):=\{f\in \nobreak K[X]\colon \text {for all}\ a\in D,\ f(xX+a)\in D[X]\}$. 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. (English) |
| Keyword:
|
Bhargava ring |
| Keyword:
|
localization |
| Keyword:
|
(locally) essential domain |
| Keyword:
|
locally free module |
| Keyword:
|
(faithfully) flat module |
| Keyword:
|
Krull dimension |
| MSC:
|
13B30 |
| MSC:
|
13C11 |
| MSC:
|
13C15 |
| MSC:
|
13F05 |
| MSC:
|
13F20 |
| idZBL:
|
Zbl 07729571 |
| idMR:
|
MR4585575 |
| DOI:
|
10.21136/MB.2022.0137-21 |
| . |
| Date available:
|
2023-05-04T17:56:45Z |
| Last updated:
|
2023-09-13 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/151683 |
| . |
| Reference:
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