# Polyadic Rings of p-Adic Integers

## Abstract

**:**

## 1. Introduction

## 2. $\left(\mathit{m},\mathit{n}\right)$-Rings of Integer Numbers from Residue Classes

**Example 1.**

## 3. Representations of p-Adic Integers

## 4. $\left(\mathit{m},\mathit{n}\right)$-Rings of p-Adic Integers

**Definition 1.**

**Definition 2.**

**Remark 1.**

**Lemma 1.**

**Proof.**

**Lemma 2.**

**Proof.**

**Theorem 1.**

**Proof.**

## 5. Conclusions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

- Neurkich, J. Algebraic Number Theory; Springer: Berlin, Germany; New York, NY, USA, 1999. [Google Scholar]
- Samuel, P. Algebraic Theory of Numbers; Hermann: Paris, France, 1972. [Google Scholar]
- Berthelot, P.; Ogus, A. Notes on Crystalline Cohomology; Princeton University Press: Princeton, NJ, USA, 1978. [Google Scholar]
- LeStum, B. Rigid Cohomology; Cambridge University Press: Cambridge, UK, 2007. [Google Scholar]
- Caruso, X. Computations with p-adic numbers. Les Cours du CIRM
**2017**, 5, 1–75. [Google Scholar] [CrossRef] - Dragovich, B.; Khrennikov, A.Y.; Kozyrev, S.V.; Volovich, I.V.; Zelenov, E.I. p-Adic mathematical physics: The first 30 years. P-Adic Num. Ultrametr. Anal. Appl.
**2017**, 9, 87–121. [Google Scholar] [CrossRef] - Vladimirov, V.S.; Volovich, I.V.; Zelenov, E.I. P-Adic Analysis And Mathematical Physics; World Scientific Publishing: Singapore, 1994; p. 319. [Google Scholar]
- Sarwar, M.; Humaira, H.; Huang, H. Fuzzy fixed point results with rational type contractions in partially ordered complex-valued metric spaces. Comment. Math.
**2018**, 58, 57–78. [Google Scholar] [CrossRef] - Zada, M.B.; Sarwar, M.; Tunc, C. Fixed point theorems in b-metric spaces and their applications to non-linear fractional differential and integral equations. J. Fixed Point Theor. Appl.
**2018**, 20, 19. [Google Scholar] [CrossRef] - Koblitz, N. p-Adic Numbers, p-Adic Analysis, and Zeta-Functions, 2nd ed.; Springer: New York, NY, USA, 1996; p. 150. [Google Scholar]
- Robert, A.M. A Course in p-Adic Analysis; Springer: New York, NY, USA, 2000. [Google Scholar] [CrossRef]
- Schikhof, W.H. Ultrametric Calculus; Cambridge University Press: Cambridge, UK, 1984. [Google Scholar]
- Duplij, S. Polyadic integer numbers and finite (m,n)-fields. P-Adic Num. Ultrametr. Anal. Appl.
**2017**, 9, 257–281. [Google Scholar] [CrossRef] [Green Version] - Duplij, S. Arity shape of polyadic algebraic structures. J. Math. Phys. Anal. Geom.
**2019**, 15, 3–56. [Google Scholar] [CrossRef] [Green Version] - Duplij, S. Polyadic Algebraic Structures; IOP Publishing: Bristol, UK, 2022; p. 461. [Google Scholar] [CrossRef]
- Gouvêa, F.Q. p-Adic Numbers. An introduction, 3rd ed.; Springer: Cham, Switzerland, 2020. [Google Scholar] [CrossRef]
- Xu, K.; Dai, Z.; Dai, Z. The formulas for the coefficients of the sum and product of p-adic integers with applications to Witt vectors. Acta Arith.
**2011**, 150, 361–384. [Google Scholar] [CrossRef] - Dörnte, W. Unterschungen über einen verallgemeinerten Gruppenbegriff. Math. Z.
**1929**, 29, 1–19. [Google Scholar] [CrossRef]

${\mathit{a}}_{\mathit{q}}$∖${\mathit{b}}_{\mathit{q}}$ | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 |
---|---|---|---|---|---|---|---|---|---|

1 | $\begin{array}{c}m=\mathit{3}\\ n=\mathit{2}\\ I=1\\ J=0\end{array}$ | $\begin{array}{c}m=\mathit{4}\\ n=\mathit{2}\\ I=1\\ J=0\end{array}$ | $\begin{array}{c}m=\mathit{5}\\ n=\mathit{2}\\ I=1\\ J=0\end{array}$ | $\begin{array}{c}m=\mathit{6}\\ n=\mathit{2}\\ I=1\\ J=0\end{array}$ | $\begin{array}{c}m=\mathit{7}\\ n=\mathit{2}\\ I=1\\ J=0\end{array}$ | $\begin{array}{c}m=\mathit{8}\\ n=\mathit{2}\\ I=1\\ J=0\end{array}$ | $\begin{array}{c}m=\mathit{9}\\ n=\mathit{2}\\ I=1\\ J=0\end{array}$ | $\begin{array}{c}m=\mathit{10}\\ n=\mathit{2}\\ I=1\\ J=0\end{array}$ | $\begin{array}{c}m=\mathit{11}\\ n=\mathit{2}\\ I=1\\ J=0\end{array}$ |

2 | $\begin{array}{c}m=\mathit{4}\\ n=\mathit{3}\\ I=2\\ J=2\end{array}$ | $\begin{array}{c}m=\mathit{6}\\ n=\mathit{5}\\ I=2\\ J=6\end{array}$ | $\begin{array}{c}m=\mathit{4}\\ n=\mathit{3}\\ I=1\\ J=1\end{array}$ | $\begin{array}{c}m=\mathit{8}\\ n=\mathit{4}\\ I=2\\ J=2\end{array}$ | $\begin{array}{c}m=\mathit{10}\\ n=\mathit{7}\\ I=2\\ J=14\end{array}$ | $\begin{array}{c}m=\mathit{6}\\ n=\mathit{5}\\ I=1\\ J=3\end{array}$ | |||

3 | $\begin{array}{c}m=\mathit{5}\\ n=\mathit{3}\\ I=3\\ J=6\end{array}$ | $\begin{array}{c}m=\mathit{6}\\ n=\mathit{5}\\ I=3\\ J=48\end{array}$ | $\begin{array}{c}m=\mathit{3}\\ n=\mathit{2}\\ I=1\\ J=1\end{array}$ | $\begin{array}{c}m=\mathit{8}\\ n=\mathit{7}\\ I=3\\ J=312\end{array}$ | $\begin{array}{c}m=\mathit{9}\\ n=\mathit{3}\\ I=3\\ J=3\end{array}$ | $\begin{array}{c}m=\mathit{11}\\ n=\mathit{5}\\ I=3\\ J=24\end{array}$ | |||

4 | $\begin{array}{c}m=\mathit{6}\\ n=\mathit{3}\\ I=4\\ J=12\end{array}$ | $\begin{array}{c}m=\mathit{4}\\ n=\mathit{2}\\ I=2\\ J=2\end{array}$ | $\begin{array}{c}m=\mathit{8}\\ n=\mathit{4}\\ I=4\\ J=36\end{array}$ | $\begin{array}{c}m=\mathit{10}\\ n=\mathit{4}\\ I=4\\ J=28\end{array}$ | $\begin{array}{c}m=\mathit{6}\\ n=\mathit{3}\\ I=2\\ J=6\end{array}$ | ||||

5 | $\begin{array}{c}m=\mathit{7}\\ n=\mathit{3}\\ I=5\\ J=20\end{array}$ | $\begin{array}{c}m=\mathit{8}\\ n=\mathit{7}\\ I=11\\ J=11,160\end{array}$ | $\begin{array}{c}m=\mathit{9}\\ n=\mathit{3}\\ I=5\\ J=15\end{array}$ | $\begin{array}{c}m=\mathit{10}\\ n=\mathit{7}\\ I=5\\ J=8680\end{array}$ | $\begin{array}{c}m=\mathit{3}\\ n=\mathit{2}\\ I=1\\ J=2\end{array}$ | ||||

6 | $\begin{array}{c}m=\mathit{8}\\ n=\mathit{3}\\ I=6\\ J=30\end{array}$ | $\begin{array}{c}m=\mathit{6}\\ n=\mathit{2}\\ I=3\\ J=3\end{array}$ | |||||||

7 | $\begin{array}{c}m=\mathit{9}\\ n=\mathit{3}\\ I=7\\ J=42\end{array}$ | $\begin{array}{c}m=\mathit{10}\\ n=\mathit{4}\\ I=7\\ J=266\end{array}$ | $\begin{array}{c}m=\mathit{11}\\ n=\mathit{5}\\ I=7\\ J=1680\end{array}$ | ||||||

8 | $\begin{array}{c}m=\mathit{10}\\ n=\mathit{3}\\ I=8\\ J=56\end{array}$ | $\begin{array}{c}m=\mathit{6}\\ n=\mathit{5}\\ I=4\\ J=3276\end{array}$ | |||||||

9 | $\begin{array}{c}m=\mathit{11}\\ n=\mathit{3}\\ I=9\\ J=72\end{array}$ |

Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. |

© 2022 by the author. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Duplij, S.
Polyadic Rings of *p*-Adic Integers. *Symmetry* **2022**, *14*, 2591.
https://doi.org/10.3390/sym14122591

**AMA Style**

Duplij S.
Polyadic Rings of *p*-Adic Integers. *Symmetry*. 2022; 14(12):2591.
https://doi.org/10.3390/sym14122591

**Chicago/Turabian Style**

Duplij, Steven.
2022. "Polyadic Rings of *p*-Adic Integers" *Symmetry* 14, no. 12: 2591.
https://doi.org/10.3390/sym14122591