Natural Numbers Set Equivalent to Ideals of Integers

This article needs to be linked to other articles. You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by adding these links. To discuss this page in more detail, feel free to use the talk page. When this work has been completed, you may remove this instance of {{MissingLinks}} from the code.

Theorem

Let $S$ be the set of all ideals of $\Z$.

Let the mapping $\psi: \N \to S$ be defined as:

$\forall b \in \N: \map \psi b = \ideal b$

where $\ideal b$ is the principal ideal of $\Z$ generated by $b$.

Then $\psi$ is a bijection.

Proof

First we show that $\psi$ is injective.

Suppose $0 < b < c$.

Then $b \in \ideal b$, but $b \notin \ideal c$, because from Principal Ideals of Integers‎, $c$ is the smallest positive integer in $\ideal c$.

Thus $\ideal b \ne \ideal c$.

It is also apparent that $b > 0 \implies \ideal b \ne \ideal 0$ as $\ideal 0 = \set 0$.

Thus $\psi$ is injective.

Surjectivity follows from Principal Ideals of Integers: every integer is the smallest strictly positive element of a principal ideal of $\Z$.

$\blacksquare$

Sources

• 1965: Seth Warner: Modern Algebra ... (previous) ... (next): Chapter $\text {IV}$: Rings and Fields: $24$. The Division Algorithm: Theorem $24.3$