Mathematica Bohemica, Vol. 148, No. 2, pp. 181-195, 2023
On Bhargava rings
Mohamed Mahmoud Chems-Eddin, Omar Ouzzaouit, Ali Tamoussit
Received September 6, 2021. Published online May 4, 2022.
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.
Affiliations: Mohamed Mahmoud Chems-Eddin, Department of Mathematics, Faculty of Sciences Dhar El Mahraz, University Sidi Mohamed Ben Abdallah, Fez, Morocco, e-mail: 2m.chemseddin@gmail.com; Omar Ouzzaouit, Higher School of Education and Training, Ibn Zohr University, Agadir, Morocco, e-mail: o.ouzzaouit@uiz.ac.ma, ouzzaouitomar@gmail.com; Ali Tamoussit, Department of Mathematics, Regional Center for Education and Training Professions Souss Massa, Inezgane, Morocco, e-mail: a.tamoussit@crmefsm.ac.ma, tamoussit2009@gmail.com
