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A Note on Quaternions over Commutative Rings

This paper extends the definition of quaternions from the real numbers to commutative unital rings, generalizing their properties and specifically analyzing the unit group within the context of halidon rings.

Original authors: Antony Telveenus

Published 2026-06-23✓ Author reviewed ⓘ

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CC BY 4.0

Original authors: Antony Telveenus

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you have a set of building blocks. For a long time, mathematicians have used a specific type of block called "Quaternions" to build complex structures, but they were only allowed to use these blocks if the ground they were standing on was made of Real Numbers (the smooth, continuous numbers we use for measuring distance, time, and weight).

This paper is like an architect saying, "What if we can build with these same blocks, but on different kinds of ground? What if the ground is made of 'integers modulo n' (like the numbers on a clock face) or other specific types of rings?"

Here is a simple breakdown of what the author, A. Telveenus, discovered:

1. The New Playground: Quaternions on "Clock" Ground

Usually, quaternions are a 4-dimensional extension of complex numbers (think of them as having a front/back, left/right, up/down, and a "time" component). They are famous because their multiplication is non-commutative.

The Analogy: Imagine putting on a pair of shoes. If you put on your left shoe first, then your right, you can walk. If you put on your right first, then your left, you can also walk. But with quaternions, the order matters! If you rotate an object "Left then Up," it ends up in a different spot than if you rotate it "Up then Left."
The Paper's Move: The author takes these 4D blocks and defines them over Commutative Rings (mathematical systems that act like clocks or remainders). Instead of infinite real numbers, the components are now limited numbers (like 0, 1, 2... up to $n-1$ ).

2. The "Magic Mirror" (The Opposite Orientation)

The paper proves that for every set of these quaternion blocks, you can build a "mirror image" version.

The Analogy: Think of a standard screw. If you turn it clockwise, it goes in. If you have a "left-handed" screw, turning it clockwise makes it go out. The paper shows that the standard quaternion ring and this "left-handed" (opposite orientation) ring are actually isomorphic.
What this means: Even though the rules for multiplying look slightly different (like a mirror reflection), the two systems are mathematically identical in structure. You can translate any problem from one to the other perfectly.

3. The "Power-Up" Formula (Calculating Powers)

One of the trickiest parts of working with these 4D blocks is calculating what happens when you multiply them by themselves many times (like $q^{100}$ ).

The Analogy: Imagine a robot that takes a step forward, then turns. If you tell it to repeat this 100 times, you don't want to simulate every single step. You want a shortcut.
The Paper's Move: The author provides a "shortcut recipe" (a recurrence formula). Instead of multiplying the blocks over and over, you can use a specific matrix (a grid of numbers) to jump straight to the answer. This makes it much easier to find the "Unit Group" (the set of blocks that can be reversed or "undone").

4. The "Ghost" Blocks (Nilpotent Elements)

In math, a "nilpotent" element is something that, if you multiply it by itself enough times, eventually vanishes into zero.

The Analogy: Imagine a special ink that fades away completely after you write with it three times. The first time it's visible, the second time it's faint, and the third time it's gone.
The Paper's Move: The author figures out exactly which quaternion blocks are these "fading inks" when working over prime number systems (like $Z_p$ ). They found a strict rule: for these blocks to vanish, their "real" part must be zero, and the sum of the squares of their other parts must also be zero. They even counted exactly how many of these "ghost" blocks exist.

5. The "Halidon" Rings: The VIP Club

This is the most technical part of the paper. A Halidon Ring is a special mathematical club. To get in, the ring must have a "primitive root of unity."

The Analogy: Think of a clock with $m$ hours. A "primitive root" is a special hand that, if you move it $m$ times, visits every single hour exactly once before returning to the start. A Halidon ring is a system where this special hand exists and where the number $m$ itself can be divided cleanly (it's invertible).
The Paper's Move:
- The author defines exactly when a ring of integers modulo $n$ qualifies as a Halidon ring.
- They created a "Halidon Function" ( $\psi$ ) that acts like a scanner. You feed it a number $n$ , and it tells you the maximum size of the "VIP club" (the index $m$ ) that ring can support.
- The Big Discovery: They proved that the ring of Quaternions over $Z_7$ (integers modulo 7) is a Halidon ring with a massive index of 48.
- The Proof: They actually listed out the powers of a specific quaternion ( $q$ ) from $q^1$ all the way to $q^{48}$ to show that it visits 48 unique states before returning to 1, and that every step in the way is "invertible" (reversible). This proves the existence of a non-commutative Halidon ring, which is a rare and valuable find.

Summary

The paper takes a complex mathematical tool (Quaternions) and successfully moves it from the smooth world of Real Numbers into the "pixelated" world of modular arithmetic (rings). It provides:

Rules for how these new blocks behave.
Shortcuts for calculating their powers.
Counting methods for special "vanishing" blocks.
Proof that these blocks can form a "Halidon Ring" (a system with a very specific, high-order cyclic structure), specifically demonstrating this with the ring of quaternions over the number 7.

The author concludes that these findings open up new connections for fields like Cryptography (making secure codes), Group Algebras (studying symmetry), and Permutation Polynomials (shuffling numbers), though the paper focuses primarily on establishing the mathematical existence and properties of these structures rather than building specific applications yet.

Technical Summary: A Note on Quaternions over Commutative Rings

Problem Statement
The classical theory of quaternions is traditionally defined over the field of real numbers ( $\mathbb{R}$ ), forming a division ring. This paper addresses the extension of this framework to a broader algebraic setting: the ring of quaternions ( $H_R$ ) defined over a commutative unital ring $R$ . The primary challenge lies in generalizing fundamental properties—such as invertibility, conjugation, and norm—when the underlying scalar system contains zero divisors and lacks the field properties of $\mathbb{R}$ . Furthermore, the paper investigates the specific structural properties of the unit group $U(H_R)$ and the behavior of quaternions over "halidon rings," a class of rings characterized by the existence of primitive roots of unity and the invertibility of their indices.

Methodology
The author employs a constructive algebraic approach to define $H_R = \{w + xi + yj + zk \mid w, x, y, z \in R\}$ with standard addition and multiplication rules derived from Hamilton's original formulation. The methodology proceeds through several stages:

Generalization of Definitions: The concepts of conjugate ( $q^*$ ) and norm ( $||q||^2 = qq^*$ ) are retained from the real case. The invertibility condition is re-evaluated in the context of ring units.
Structural Isomorphism: The paper constructs an "induced" ring of quaternions ( $H^\perp_R$ ) with an opposite orientation (swapping the multiplication rules for $i, j, k$ ) and proves its isomorphism to the standard $H_R$ via a specific bijective map.
Recursive Computation: Mathematical induction and matrix representations are utilized to derive recurrence formulas for computing powers of quaternions ( $q^n$ ) and to characterize the unit group $U(H_R)$ for finite commutative rings.
Case Studies on Specific Rings: The analysis focuses on rings of integers modulo $n$ ( $\mathbb{Z}_n$ ), specifically examining nilpotent elements, tripotents, and the conditions under which $\mathbb{Z}_n$ constitutes a halidon ring.
Halidon Ring Integration: The paper integrates the theory of halidon rings (rings possessing a primitive $m$ -th root of unity $\omega$ where $m$ is invertible) with quaternion algebras, establishing necessary and sufficient conditions for $\mathbb{Z}_n$ to be a halidon ring and calculating the number of primitive roots within these structures.

Key Contributions and Results

Invertibility Criterion: The paper establishes that an element $q \in H_R$ is invertible if and only if its squared norm $||q||^2$ is a unit in the base ring $R$ (Proposition 1).
Orientation Isomorphism: It is proven that the ring of quaternions with standard orientation is isomorphic to a ring with opposite orientation ( $H_R \cong H^\perp_R$ ), demonstrating that the choice of orientation does not alter the ring structure (Theorem 2).
Power Recurrence Relations: A recurrence formula is derived for $q^n$ where $q = w + u$ and $||u||=1$ , expressing the power in terms of scalar and vector components using a specific $2 \times 2$ matrix transformation (Theorem 3).
Unit Group Characterization: For finite commutative rings, the unit group $U(H_R)$ is characterized as the set of elements $q$ such that $q^s = 1$ for some integer $s$ , with a matrix-based recurrence provided for computing these powers (Theorem 4).
Nilpotent Elements in $\mathbb{Z}_p$ : For primes $p > 3$ , the paper characterizes nilpotent elements in $H_{\mathbb{Z}_p}$ as those with a zero scalar part ( $w=0$ ) and a vector part satisfying $x^2 + y^2 + z^2 = 0$ . The index of nilpotency is shown to be 2, and the total count of such elements is $p^2$ (Theorem 6).
Tripotent Properties: Proposition 7 details the properties of tripotent elements ( $q^3=q$ ), noting that genuine tripotents cannot have a zero scalar part and that $q^2$ and $1-q^2$ form orthogonal idempotents leading to a Pierce decomposition.
Halidon Rings and Quaternions: The paper provides a complete characterization of $\mathbb{Z}_n$ as a halidon ring based on the prime factorization of $n$ and the greatest common divisor of Euler's totient functions of its prime factors (Theorem 13, Proposition 14). It defines a "halidon function" $\psi(n)$ representing the maximum index of the ring.
Non-Commutative Halidon Example: A concrete example is provided where $H_{\mathbb{Z}_7}$ is shown to be a non-commutative halidon ring with a maximum index of 48, containing a primitive 48th root of unity (Example 2).

Significance and Claims
The paper claims to extend the foundational theory of quaternions from fields to commutative unital rings, thereby creating a "rich class of non-commutative halidon rings." The author asserts that this work broadens the scope of mathematical research by establishing connections between quaternion algebras and halidon rings.

The significance is framed around potential applications in:

Cryptography: Utilizing these algebraic structures for the design and analysis of secure systems.
Group Algebras: Offering new avenues for theoretical investigation through quaternion-based ring structures.
Permutation Polynomials: Providing a broader setting for studying polynomial behavior over halidon rings.

The paper concludes modestly, stating that these results create "useful connections across several fields" without claiming to solve specific open problems in those fields, but rather by providing the algebraic infrastructure (the rings and their properties) necessary for such investigations.