Definition:Principal Ideal of Ring

Definition
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 $$a$$.

From Principal Ideal is an Ideal, $$\left({a}\right)$$ is a principal ideal if $$\left \langle {a} \right \rangle$$ is the ideal generated by $$a$$.