Definition:Principal Ideal of Ring

Theorem
Let $$\left({R, +, \circ}\right)$$ be a ring with unity.

Let $$a \in R$$.

We define $$\left({a}\right) = \left\{{r \circ a: r \in R}\right\}$$.

Then:
 * 1) $$\forall a \in R: \left({a}\right)$$ is an ideal of $$R$$;
 * 2) $$\forall a \in R: a \in \left({a}\right)$$;
 * 3) $$\forall a \in R:$$ if $$J$$ is an ideal of $$R$$, and $$a \in J$$, then $$\left({a}\right) \subseteq J$$. That is, $$\left({a}\right)$$ is the smallest ideal of $$R$$ containing $$a$$.

The ideal $$\left({a}\right)$$ is called the principal ideal of $$R$$ generated by $$a$$.

That is, an ideal is a principal ideal if $$\exists a \in R$$ such that $$\left \langle {a} \right \rangle$$ is the ideal generated by $$a$$.

Proof
Let $$a \in R$$.

First we establish that $$\left({a}\right)$$ is an ideal of $$R$$, by verifying the conditions of Test for Ideal.


 * $$a \ne \varnothing$$, as $$1_R \circ a = a \in \left({a}\right)$$.
 * Let $$x, y \in \left({a}\right)$$. Then:

$$ $$ $$ $$


 * Let $$s \in \left({a}\right), x \in R$$.

$$ $$ $$ $$

... and similarly $$s \circ x \in \left({a}\right)$$.

Thus by Test for Ideal, $$\left({a}\right)$$ is an ideal of $$R$$.


 * Now let $$J$$ be an ideal of $$R$$ such that $$a \in J$$.

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

So every element of $$\left({a}\right)$$ is in $$J$$, thus $$\left({a}\right) \subseteq J$$.