# Recapturing the Structure of Group of Units of Any Finite Commutative Chain Ring

^{*}

^{†}

## Abstract

**:**

## 1. Introduction

## 2. Preliminaries and Notations

**Proposition**

**1**

## 3. The Determination of the Structure of $\mathit{H}$

**Proposition**

**2.**

**Proof.**

**Definition**

**1.**

**Corollary**

**1.**

**Remark**

**1.**

**Proposition**

**3.**

**Theorem**

**1.**

**Proof.**

**Corollary**

**2.**

**Corollary**

**3.**

**Remark**

**2.**

- (1)
- (2)
- If $p\nmid k,$ then obviously $\lambda =1\le \mu $ and, thus, the result that is given in [7] is just a particular case of Theorem 1.

**Lemma**

**1.**

**Proof.**

**Remark**

**3.**

**Theorem**

**2.**

- (i)
- ${U}_{s}$ is a direct product of r cyclic groups of order ${p}^{\nu \left(s\right)}.$
- (ii)
- ${U}_{{s}_{i}}^{*}$ is a direct product of $r-1$ cyclic groups of order ${p}^{\nu \left({s}_{i}\right)},$ and ${D}_{i}$ is a cyclic group of order ${p}^{{\mu}_{i}}.$
- (iii)
- D is a cyclic group of order ${\lambda}_{0}=\lambda +{\sum}_{i=1}^{d}(\nu \left({s}_{i}\right)-{\mu}_{i}).$
- (iv)
- C is a cyclic group of order ${p}^{\nu \left(u\right)-1}$ if $({i}^{\prime},{s}^{\prime})\notin I$ and $C=<1>$ otherwise.

**Proof.**

**Corollary**

**4.**

**Corollary**

**5.**

**Example**

**1.**

**Corollary**

**6.**

**Example**

**2.**

## 4. Enumeration of Generators of the Same Order

**Theorem**

**3.**

**Proof.**

**Corollary**

**7.**

**Proposition**

**4.**

**Proof.**

**Remark**

**4.**

**Theorem**

**4.**

**Proof.**

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**MDPI and ACS Style**

Alabiad, S.; Alkhamees, Y.
Recapturing the Structure of Group of Units of Any Finite Commutative Chain Ring. *Symmetry* **2021**, *13*, 307.
https://doi.org/10.3390/sym13020307

**AMA Style**

Alabiad S, Alkhamees Y.
Recapturing the Structure of Group of Units of Any Finite Commutative Chain Ring. *Symmetry*. 2021; 13(2):307.
https://doi.org/10.3390/sym13020307

**Chicago/Turabian Style**

Alabiad, Sami, and Yousef Alkhamees.
2021. "Recapturing the Structure of Group of Units of Any Finite Commutative Chain Ring" *Symmetry* 13, no. 2: 307.
https://doi.org/10.3390/sym13020307