Zero Divides Zero

Theorem

Let $n \in \Z$ be an integer.

Then:

$0 \divides n \implies n = 0$

That is, zero is the only integer divisible by zero.

$\ds 0$

$\divides$

$\ds n$

$\ds \leadsto \ \ $

$\ds \exists q \in \Z: \, $

$\ds n$

$=$

$\ds q \times 0$

Definition of Divisor of Integer

$\ds \leadsto \ \ $

$\ds n$

$=$

$\ds 0$

Integers have no zero divisors, as Integers form Integral Domain.

$\blacksquare$

1969: C.R.J. Clapham: Introduction to Abstract Algebra ... (previous) ... (next): Chapter $3$: The Integers: $\S 10$. Divisibility
1982: P.M. Cohn: Algebra Volume 1 (2nd ed.) ... (previous) ... (next): Chapter $2$: Integers and natural numbers: $\S 2.2$: Divisibility and factorization in $\mathbf Z$
1994: H.E. Rose: A Course in Number Theory (2nd ed.) ... (previous) ... (next): $1$ Divisibility: $1.1$ The Euclidean algorithm and unique factorization: $\text {(ii)}$