Unit Ideal is Principal Ideal Generated by Unity
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Theorem
Let $A$ be a commutative ring with unity.
Then:
- $A = \ideal 1$
where:
- $A$ is called the unit ideal of $A$
- $\ideal 1$ denotes the principal ideal generated by the unity of $A$
Proof
$\ideal 1 \subseteq A$ is clear by definition of principal ideal.
To see $A \subseteq \ideal 1$, let $a \in A$ be an arbitrary element.
Then:
\(\ds a\) | \(=\) | \(\ds a 1\) | Definition of Identity Element | |||||||||||
\(\ds \) | \(\in\) | \(\ds \ideal 1\) | Definition of Principal Ideal of Ring |
$\blacksquare$