Principal Ideals of Integers

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Theorem

Let $J$ be a non-zero ideal of $\Z$.

Then $J = \ideal b$ where $b$ is the smallest strictly positive integer belonging to $J$.


Proof

It follows from Ring of Integers is Principal Ideal Domain that $J$ is a principal ideal.


Let $c \in J, c \ne 0$.

Then $-c \in J$ and by Natural Numbers are Non-Negative Integers, exactly one of them is strictly positive.

Thus $J$ does actually contain strictly positive elements, so that's a start.


Let $b$ be the smallest strictly positive element of $J$.

This exists because Natural Numbers are Non-Negative Integers and the Well-Ordering Principle.

By definition of a principal ideal, we have $\ideal b \subseteq J$ as $b \in J$.

We need to show that $J \subseteq \ideal b$.

So, let $a \in J$.

By the Division Theorem, $\exists q, r: a = b q + r, 0 \le r < b$.

As $a, b \in J$, then so does $r = a - b q$.

So, by the definition of $b$, it follows that $r = 0$.

Thus $a = b q \in \ideal b$.

$\blacksquare$


Sources

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