Non-Field Integral Domain has Infinite Number of Ideals

Theorem

Let $\struct {D, +, \circ}$ be an integral domain which is not a field.

Then $\struct {D, +, \circ}$ has an infinite number of distinct ideals.


Proof

Let $a \in D$ be a proper element of $D$.

Because $\struct {D, +, \circ}$ is not a field, such an element is known to exist.


From Principal Ideal in Integral Domain generated by Power Plus One is Subset of Principal Ideal generated by Power:

$\forall n \in \Z_{\ge 0}: \ideal {a^{n + 1} } \subsetneq \ideal {a_n}$

where $\ideal x$ denotes the principal ideal of $D$ generated by $x$.


Hence the set:

$S = \set {\ideal {1_D}, \ideal a, \ideal {a^2}, \ideal {a^3}, \dotsc}$

is infinite.

$\blacksquare$


Sources

  • 1978: Thomas A. Whitelaw: An Introduction to Abstract Algebra ... (previous) ... (next): Chapter $9$: Rings: Exercise $7 \ \text {(i)}$
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