Principal Ideal of Commutative Ring

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

Let $a \in R$.

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

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

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

Then $Ra = \ideal a = aR$.

Proof
By definition of principal left ideal:
 * $Ra = {r \circ a: r \in R}$

By definition of commutative ring with unity and center of ring:
 * $a$ is in the center of $R$

From Principal Ideal from Element in Center of Ring:
 * $\ideal a = R \circ a = {r \circ a: r \in R}$

Hence:
 * $Ra = \ideal a$

We have: