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Free nearrings

This paper provides explicit constructions for free nearrings and the free product of two nearrings within the variety of all nearrings, including those that are not necessarily zero-symmetric.

Original authors: Stefan Veldsman

Published 2026-08-11

📖 8 min read🧠 Deep dive

CC0 1.0

Original authors: Stefan Veldsman

Original paper dedicated to the public domain under CC0 1.0 (http://creativecommons.org/publicdomain/zero/1.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine a world where math isn't just about numbers on a page, but about the rules that govern how things combine, stack, and interact. This is the realm of abstract algebra, a branch of mathematics that studies structures like groups and rings. Think of a "group" as a club with a specific set of members and a single rule for how they can combine (like adding numbers). A "ring" is a more complex club with two rules: one for adding and one for multiplying. But what if we loosen the rules just a tiny bit? What if multiplication only plays nice when it comes from the right side, but ignores the left? That's the quirky, slightly rebellious world of a "nearring."

Nearrings are like the wild cousins of rings. In a standard ring, multiplication distributes over addition perfectly on both sides (just like $a(b+c) = ab + ac$ and $(a+b)c = ac + bc$). In a nearring, we only demand that it works from the right: $(a+b)c = ac + bc$. The left side? It can do whatever it wants. This flexibility makes nearrings incredibly useful for modeling real-world systems where order and direction matter, like in computer science or geometry. But because they are so flexible, they are also notoriously hard to build from scratch. Mathematicians often want to know: "If I start with a blank slate and a few basic ingredients, what is the most general, rule-abiding nearring I can create?" This is the quest for a "free nearring."

This paper by Stefan Veldsman tackles that exact question. The author provides a detailed, step-by-step recipe for constructing a "free nearring" from a simple set of symbols, as well as a method for merging two existing nearrings into a new, combined one (called a "free product"). While previous work had only solved this puzzle for a very specific, well-behaved type of nearring (one where zero acts nicely), Veldsman proves that this construction works for all nearrings, even the messy, unpredictable ones where zero doesn't behave perfectly. The paper doesn't just suggest this is possible; it offers a rigorous, mathematical proof that these structures can be built exactly as described, expanding the toolkit available to mathematicians working with these complex algebraic systems.

The Lego Set of Math: Building a Free Nearring

Imagine you are a master builder with a box of Lego bricks. You want to build the ultimate, most flexible castle possible, but you have a very specific rule: every time you snap a new piece onto the right side of a tower, the whole structure must stay stable. However, if you snap a piece on the left, the tower might wobble, tilt, or do something weird. This is the world of nearrings.

In this paper, the author, Stefan Veldsman, acts as the architect. He wants to show us how to build the "Free Nearring." Think of a "free" structure in math as the ultimate blank canvas. It's the most basic, unadorned version of a mathematical object you can make using a specific set of starting ingredients (like a set of letters, $X$ ). It contains only the rules you forced it to have and nothing else. If you want to build a nearring, you need to know how to build this "free" version first, because every other nearring is just a simplified copy of it.

The Ingredients: A Semigroup with a "Left Zero"

Before we can build the nearring, we need a foundation. Veldsman starts with something called a semigroup with a left zero.

The Semigroup: Imagine a line of people passing a ball. The rule is simple: if Person A passes to Person B, and Person B passes to Person C, it's the same as Person A passing directly to Person C. It's just a chain of actions.
The Left Zero: Now, imagine there is a special person in this line called "Zero." The rule for Zero is strict: if Zero passes the ball to anyone, the ball disappears (or stays Zero). But if anyone passes the ball to Zero, Zero keeps it and passes it back to them unchanged. It's a "left zero" because the magic happens when Zero is on the left side of the interaction.

Veldsman constructs a specific set of these "words" or chains of symbols (like $x_1x_2x_3$ ) and adds this special Zero element. This forms the raw material, the "R," from which the nearring will be built.

The Construction: A Tower of Groups

Building the nearring isn't a one-step process; it's like building a skyscraper floor by floor. The author uses a method called induction, which means he builds the first floor, then uses that to build the second, and so on, forever.

The Ground Floor ( $B_0$ ): He starts by taking the raw materials (the semigroup $R$ ) and turning them into a free additive group. Think of this as taking all your Lego bricks and allowing yourself to add them together or subtract them (like having a "negative" brick). This creates a massive, flexible group where you can combine elements in any order you like, as long as you follow the basic rules of addition.
The Next Floors ( $B_1, B_2, \dots$ ): Here is where the magic of the nearring happens. The author introduces a new layer of complexity. He takes the current floor and combines it with a new set of "formal products" (imagine gluing a new type of brick onto the existing structure).
- He creates a new set of elements called $D_n$ , which are essentially the old elements multiplied by the new "free" elements.
- He then builds the next floor ( $B_n$ ) by taking the previous floor and "free producting" it with the new elements. In math-speak, a "free product" is like gluing two separate groups together at a single point, creating a giant, branching tree of possibilities.
- He repeats this process infinitely. The final "Free Nearring" is the union of all these floors, a towering structure of infinite complexity.

The Rules of the Game: Distributivity and Associativity

Once the tower is built, Veldsman has to prove it actually works as a nearring. He checks two critical rules:

Right Distributivity: This is the golden rule of nearrings. It says that if you have a sum $(a + b)$ and you multiply it by $c$ on the right, it must equal $ac + bc$. Veldsman shows that because of the way he built the tower (floor by floor, respecting the rules at every step), this rule holds true perfectly. The structure is designed so that multiplication from the right always plays nice with addition.
Associativity: This rule says that $(ab)c$ must equal $a(bc)$. It's the rule that says it doesn't matter how you group your operations. Veldsman proves this by showing that no matter which "floor" of the tower your elements live on, the multiplication works consistently.

The "Free Product" of Two Nearrings

The paper also tackles a second challenge: What happens if you have two existing nearrings, say $A$ and $B$ , and you want to smash them together into one giant nearring without losing any of their original identity? This is called the free product (or coproduct).

Veldsman uses a similar "tower-building" strategy. He starts by taking the "multiplicative" parts of $A$ and $B$ and merging them into a new semigroup $R$ . He then runs the same infinite construction process. The result is a new nearring that contains both $A$ and $B$ as sub-parts. If you have a rule in $A$ or a rule in $B$ , it stays intact in the new giant structure. The only new rules added are the bare minimum required to make them work together as a single nearring.

Why This Matters

You might wonder, "Who cares about building these abstract towers?" The answer lies in the flexibility. Nearrings are used to model things where direction matters, like functions in computer science or transformations in geometry. By having a precise, "free" blueprint for how to build these structures, mathematicians can:

Test Theories: They can use the free nearring as a "stress test" to see if a new mathematical idea holds up under the most extreme conditions.
Understand Complexity: It helps them understand the fundamental limits of what a nearring can be.
Generalize: This paper is a big deal because it removes a major restriction. Previous work only worked for "zero-symmetric" nearrings (where $a \times 0 = 0$ ). Veldsman shows that the construction works even when $a \times 0$ is not zero, opening the door to a much wider variety of mathematical models.

In short, Stefan Veldsman has handed us the master key to the nearring universe. He didn't just guess how to build it; he provided a rigorous, step-by-step proof that these structures exist, can be built, and follow the rules we expect them to. It's a bit like finally figuring out the exact instructions to build a Lego castle that can survive any earthquake, no matter how the bricks are stacked.

Technical Summary: Free Nearrings and Free Products in the Variety of All Nearrings

Problem Statement
The paper addresses the construction of free objects within the variety of all right nearrings, specifically extending previous work that was limited to the variety of zero-symmetric left nearrings. A nearring $N = (N, +, \cdot, 0_N)$ is defined as a structure where $(N, +, 0_N)$ is a group (not necessarily commutative), $(N, \cdot)$ is a semigroup, and multiplication distributes from the right over addition ($(a+b)c = ac + bc$). Unlike rings, nearrings do not necessarily satisfy $0a = 0$ or $a(-b) = -(ab)$; those that do are termed "zero-symmetric." The paper notes that any nearring decomposes into a zero-symmetric part and a constant part. While Meldrum [5] had previously constructed free nearrings and free products for zero-symmetric left nearrings, the explicit construction for the broader variety of all right nearrings (including non-zero-symmetric ones) remained to be detailed.

Methodology
The author, Stefan Veldsman, employs an inductive construction method, largely following the framework established by Meldrum but adapting it to handle right nearrings and the presence of constant elements. The construction proceeds in two main phases:

Construction of the Free Nearring $F(X)$ :
- Base Semigroup: The process begins with a free multiplicative semigroup with a left zero, denoted $(R, \cdot, 0_R)$ , over a generating set $X$ . This set $R$ consists of words over $X$ and formal products involving a left zero symbol $0_R$ .
- Inductive Group Tower: A sequence of groups $B_n$ is constructed. $B_0$ is the free additive group on $R$ . Subsequent groups $B_n$ are formed as the free product of the previous group $B_{n-1}$ and the free additive group on a set of formal products $D_{n-1} = R \cdot C_{n-1}$ , where $C_{n-1}$ represents the "new" elements added at the previous step.
- Limit Object: The free nearring $F(X)$ is defined as the union of this tower of groups ( $\bigcup_{n=0}^{\infty} B_n$ ).
- Multiplication Definition: Multiplication is defined inductively. It is first defined for elements in $R$ acting on $F(X)$ , then extended to additive inverses, and finally to the entire group $B_0$ and subsequent $B_n$ via distributivity and formal products.
- Verification: The author rigorously verifies that the resulting structure satisfies the nearring axioms: right distributivity, associativity of multiplication, and the existence of a well-defined product.
Construction of the Free Product (Coproduct):
- The free product of two nearrings $A$ and $B$ is constructed by first forming the free product of their underlying multiplicative semigroups with left zeros, denoted $R$ .
- The same inductive group tower and multiplication extension steps used for $F(X)$ are applied to this new base $R$ to generate the coproduct in the category of nearrings.

Key Contributions and Results

Explicit Construction: The paper provides an explicit, step-by-step construction of the free nearring $F(X)$ on a set $X$ within the variety of all right nearrings. This includes a detailed description of the underlying sets, the inductive definition of the additive group structure, and the recursive definition of the multiplication operation.
Generalization: The work extends Meldrum's results from zero-symmetric left nearrings to the general variety of right nearrings. A specific focus is placed on managing "constant elements" (elements where $ab=a$ for all $b$ ) and non-zero-symmetric behavior, which were not the primary focus of the earlier zero-symmetric constructions.
Free Product (Coproduct): The paper successfully constructs the free product of two arbitrary nearrings $A$ and $B$ . It demonstrates that the resulting structure $F(R)$ (where $R$ is the semigroup coproduct of $A$ and $B$ ) satisfies the universal property of a coproduct in the category of nearrings.
Associativity and Distributivity: The author provides detailed proofs that the constructed multiplication is associative and right-distributive over addition, even in the presence of reductions in the additive group words.
Unital Extensions: The paper outlines how to modify the construction to produce free nearrings with an identity element and how to construct the free product of a nearring with the ring of integers $\mathbb{Z}$ , effectively adjoining an identity.

Significance and Claims
The paper claims to fill a gap in the literature by providing the necessary details for constructing free objects in the variety of all nearrings, not just the zero-symmetric subvariety. The author notes that while one could theoretically follow Meldrum's presentation "almost verbatim," the specific handling of constant elements and the right-distributive nature of the variety requires the additional details provided herein.

The significance lies in establishing the existence and explicit form of free objects in this broader category, which is a fundamental requirement for understanding the structural properties of nearrings. The paper does not propose new applications or experimental uses but rather solidifies the theoretical foundation for free nearrings and their coproducts. As noted in the introduction, the construction relies on the universal properties of free groups and semigroups, adapted to the specific algebraic constraints of nearrings. The work serves as a technical extension of Meldrum's [5] and [4] foundational papers, ensuring the theory covers the full variety of right nearrings.