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...
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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 |
| _version_ | 1797837610080534528 |
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| author | Mohamed Mahmoud Chems-Eddin Omar Ouzzaouit Ali Tamoussit |
| author_facet | Mohamed Mahmoud Chems-Eddin Omar Ouzzaouit Ali Tamoussit |
| author_sort | Mohamed Mahmoud Chems-Eddin |
| collection | DOAJ |
| description | 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. |
| first_indexed | 2024-04-09T15:28:35Z |
| format | Article |
| id | doaj.art-f9f12ae3d6b54c509c2c674a56c7584b |
| institution | Directory Open Access Journal |
| issn | 0862-7959 2464-7136 |
| language | English |
| last_indexed | 2024-04-09T15:28:35Z |
| publishDate | 2023-07-01 |
| publisher | Institute of Mathematics of the Czech Academy of Science |
| record_format | Article |
| series | Mathematica Bohemica |
| spelling | doaj.art-f9f12ae3d6b54c509c2c674a56c7584b2023-04-28T12:19:40ZengInstitute of Mathematics of the Czech Academy of ScienceMathematica Bohemica0862-79592464-71362023-07-01148218119510.21136/MB.2022.0137-21MB.2022.0137-21On Bhargava ringsMohamed Mahmoud Chems-EddinOmar OuzzaouitAli TamoussitLet $D$ 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.http://mb.math.cas.cz/full/148/2/mb148_2_3.pdf bhargava ring localization (locally) essential domain locally free module (faithfully) flat module krull dimension |
| spellingShingle | Mohamed Mahmoud Chems-Eddin Omar Ouzzaouit Ali Tamoussit On Bhargava rings Mathematica Bohemica bhargava ring localization (locally) essential domain locally free module (faithfully) flat module krull dimension |
| title | On Bhargava rings |
| title_full | On Bhargava rings |
| title_fullStr | On Bhargava rings |
| title_full_unstemmed | On Bhargava rings |
| title_short | On Bhargava rings |
| title_sort | on bhargava rings |
| topic | bhargava ring localization (locally) essential domain locally free module (faithfully) flat module krull dimension |
| url | http://mb.math.cas.cz/full/148/2/mb148_2_3.pdf |
| work_keys_str_mv | AT mohamedmahmoudchemseddin onbhargavarings AT omarouzzaouit onbhargavarings AT alitamoussit onbhargavarings |