Is There an Infinite Commutative Ring with Unity That Has a Finite Set of Zero Divisors?

This article explores the properties of infinite commutative rings with a unity element, focusing on the specific case where the number of zero divisors is finite. The analysis combines theoretical insights with practical examples to clarify the conditions under which such rings can exist.

Introduction

The question we aim to address is whether there exists an infinite commutative ring with a unity element (a unit or 1) that possesses a finite set of zero divisors. This exploration is crucial for understanding the structure and behavior of algebraic structures in abstract algebra.

Key Concepts and Definitions

Before diving into the proof, it is essential to define and establish key concepts:

Commutative Ring with Unity: A ring in which the multiplication operation is commutative and there exists a multiplicative identity element, commonly denoted as 1. Zero Divisor: An element (a) in a ring (R) is a zero divisor if there exists a non-zero element (b) in (R) such that (a cdot b 0) or (b cdot a 0).

Our goal is to demonstrate that, in an infinite commutative ring with unity, the set of zero divisors is necessarily either empty or infinite, with the only exception being the trivial ring where the only zero divisor is 0.

The Proof

The argument proceeds as follows:

Let (R) be an infinite commutative ring with unity. Assume, for the sake of contradiction, that (R) has a finite number of zero divisors, including 0. Denote the zero divisors in (R) as (a_0 0, a_1, a_2, ldots, a_k), where (k) is a non-negative integer representing the number of non-zero zero divisors in (R).

Case Analysis: Only the Trivial Zero Divisor

The simplest case is when the set of zero divisors is just (0). In this scenario, it is straightforward because every element in (R) is non-zero and non-divisors of 0. This case is trivial and aligns with the ring (mathbb{Z}), the ring of integers, which has no non-zero zero divisors.

General Case: Infinitely Many Zero Divisors

For the general case, assume there are at least (2) non-zero zero divisors in (R). Let (a_1) and (a_2) be two such non-zero zero divisors. By definition, there exist non-zero elements (y_1, y_2 in R) such that (y_1 cdot a_1 0) and (y_2 cdot a_2 0).

Consider any (r in R). Using the associativity property of multiplication, we have:

[ y_1 cdot (r cdot a_1) (y_1 cdot r) cdot a_1 0 cdot a_1 0 ]

Since (a_1) is a zero divisor, (r cdot a_1) must be a zero divisor in (R). This means (r cdot a_1 a_i) for some (0 leq i leq k).

Assume (R) has at least (k cdot k_3) elements, denoted as (r_1, r_2, ldots, r_{k cdot k_3}). By the pigeonhole principle, since (a_1 cdot r_j) can only take on one of (k) possible values (the zero divisors including 0), at least (k_3) of the (r_j) must map to the same zero divisor (a_i).

Without loss of generality, suppose:

[ a_1 cdot r_1 a_1 cdot r_2 ldots a_1 cdot r_{k_3} a_i ]

From this, we get:

[ r_2 - r_1 r_3 - r_1 ldots r_{k_3} - r_1 0 ]

This implies that all these differences are equal to 0, leading to a configuration where at least (k_3) elements of (R) are identical, contradicting the assumption that (R) is infinite.

Conclusion

Through this proof by contradiction, we establish that an infinite commutative ring with a unity element cannot have a finite number of zero divisors. The only exception is the trivial case where the only zero divisor is 0, as illustrated by the ring (mathbb{Z}).

Understanding these properties is vital for various applications in algebra and theoretical computer science, emphasizing the importance of the structure and constraints on infinite commutative rings.