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 $R$.

Also, if $J$ is an ideal of $R$ and $a \in J$, then $\ideal a \subseteq J$.

That is, $\ideal a$ is the smallest ideal of $R$ to which $a$ belongs.

Proof of Ideal
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 $R$.

Proof that Principal Ideal is Smallest
Let $J$ be an ideal of $R$ such that $a \in J$.

By the definition of an ideal:
 * $\forall r, s \in R: r \circ a \circ s \in J$

Also, $J$ is a group under $+$.

So every element of $\ideal a$ is in $J$.

Thus $\ideal a \subseteq J$.