Unit Ideal is Principal Ideal Generated by Unity

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

$\blacksquare$