Principal Ideal is Ideal

Theorem
Let $\struct {R, +, \circ}$ be a ring with unity.

Let $a \in R$.

Let $\ideal a$ be the principal ideal of $R$ generated by $a$.

Then $\ideal a$ is an ideal of $R$.

Proof
First we establish that $\ideal a$ is an ideal of $R$, by verifying the conditions of Test for Ideal.

$\ideal a \ne \O$, as $1_R \circ a = a \in \ideal a$.

Let $x, y \in \ideal a$.

Then:

Let $s \in \ideal a, x \in R$.

and similarly $s \circ x \in \ideal a$.

Thus by Test for Ideal, $\ideal a$ is an ideal of $R$.