Principal Ideal Domain cannot have Infinite Strictly Increasing Sequence of Ideals

Theorem

Let $\struct {D, +, \circ}$ be a principal ideal domain.

Then $D$ cannot have an infinite sequence of ideals $\sequence {j_n}_{n \mathop \in \N}$ such that:

$\forall n \in \N: J_n \subsetneq j_{n + 1}$

Proof

Let $K = \ds \bigcup_{n \mathop \in \N} J_n$.

Then from Increasing Union of Sequence of Ideals is Ideal, $K$ is an ideal of $D$.

We have that $D$ is a principal ideal domain.

Hence there exists $a \in D$ such that:

$K = \ideal a$

where $\ideal a$ is the principal ideal of $D$ generated by $a$.

But $a \in J_m$ for some $m \in \N$.

Thus $K \subseteq J_m$

Thus it follows that $J_{m + 1} \subseteq J_m$ which contradicts our initial assertion that:

$\forall n \in \N: J_n \subsetneq j_{n + 1}$

Hence the result.

$\blacksquare$

Sources

• 1978: Thomas A. Whitelaw: An Introduction to Abstract Algebra ... (previous) ... (next): Chapter $9$: Rings: Exercise $17$