# Ring of Integers has no Zero Divisors

## Theorem

The integers have no zero divisors:

- $\forall x, y, \in \Z: x \times y = 0 \implies x = 0 \lor y = 0$

This can equivalently be expressed:

- $\forall x, y, \in \Z: x \ne 0 \land y \ne 0 \implies x \times y \ne 0$

## Proof

Let us define $\eqclass {\tuple {a, b} } \boxminus$ as in the formal definition of integers.

That is, $\eqclass {\tuple {a, b} } \boxminus$ is an equivalence class of ordered pairs of natural numbers under the congruence relation $\boxminus$.

$\boxminus$ is the congruence relation defined on $\N \times \N$ by $\tuple {x_1, y_1} \boxminus \tuple {x_2, y_2} \iff x_1 + y_2 = x_2 + y_1$.

In order to streamline the notation, we will use $\eqclass {a, b} {}$ to mean $\eqclass {\tuple {a, b} } \boxminus$, as suggested.

From the method of construction, $\eqclass {c, c} {}$, where $c$ is any element of the natural numbers $\N$, is the identity of $\struct {\Z, +}$.

To ease the algebra, we will take $\eqclass {0, 0} {}$ as a canonical instance of this equivalence class.

We need to show that:

- $\forall a, b, c, d \in \N: \eqclass {a, b} {} \times \eqclass {c, d} {} = \eqclass {0, 0} {} \implies \eqclass {a, b} {} = \eqclass {0, 0} {} \lor \eqclass {c, d} {} = \eqclass {0, 0} {}$

From Natural Numbers form Commutative Semiring, we can take it for granted that:

- addition and multiplication are commutative and associative on the natural numbers $\N$
- natural number multiplication is distributive over natural number addition.

So:

\(\displaystyle \eqclass {a, b} {} \times \eqclass {c, d} {}\) | \(=\) | \(\displaystyle \eqclass {0, 0} {}\) | |||||||||||

\(\displaystyle \leadsto \ \ \) | \(\displaystyle \eqclass {a c + b d, a d + b c} {}\) | \(=\) | \(\displaystyle \eqclass {0, 0} {}\) | ||||||||||

\(\displaystyle \leadsto \ \ \) | \(\displaystyle a c + b d + 0\) | \(=\) | \(\displaystyle a d + b c + 0\) | ||||||||||

\(\displaystyle \leadsto \ \ \) | \(\displaystyle a c + b d\) | \(=\) | \(\displaystyle a d + b c\) |

We have to be careful here, and bear in mind that $a, b, c, d$ are natural numbers, and we have not defined (and, at this stage, will not define) subtraction on such entities.

Without loss of generality, suppose that $\eqclass {c, d} {} \ne \eqclass {0, 0} {}$.

Then $c \ne d$.

Without loss of generality, suppose also that $c > d$.

From Ordering in terms of Addition, $\exists p \in \N: d + p = c$ where $p > 0$.

Then:

\(\displaystyle a c + b d\) | \(=\) | \(\displaystyle a d + b c\) | |||||||||||

\(\displaystyle \leadsto \ \ \) | \(\displaystyle a \paren {d + p} + b d\) | \(=\) | \(\displaystyle a d + b \paren {d + p}\) | ||||||||||

\(\displaystyle \leadsto \ \ \) | \(\displaystyle a d + a p + b d\) | \(=\) | \(\displaystyle a d + b d + b p\) | ||||||||||

\(\displaystyle \leadsto \ \ \) | \(\displaystyle a p\) | \(=\) | \(\displaystyle b p\) | ||||||||||

\(\displaystyle \leadsto \ \ \) | \(\displaystyle a\) | \(=\) | \(\displaystyle b\) | ||||||||||

\(\displaystyle \leadsto \ \ \) | \(\displaystyle \eqclass {a, b} {}\) | \(=\) | \(\displaystyle \eqclass {0, 0} {}\) | Construction of Inverse Completion: Equivalence Class of Equal Elements |

Similarly for when $c < d$.

Thus:

- $\eqclass {c, d} {} \ne \eqclass {0, 0} {} \implies \eqclass {a, b} {} = \eqclass {0, 0} {}$

A similar argument shows that:

- $\eqclass {a, b} {} \ne \eqclass {0, 0} {} \implies \eqclass {c, d} {} = \eqclass {0, 0} {}$

The equivalence between the two forms of the statement of this theorem follows from De Morgan's Laws: Conjunction of Negations and the Rule of Transposition.

$\blacksquare$

## Sources

- 1982: P.M. Cohn:
*Algebra Volume 1*(2nd ed.) ... (previous) ... (next): Chapter $2$: Integers and natural numbers: $\S 2.1$: The integers: $\mathbf Z. \, 6$