Principal Right Ideal is Right Ideal

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

Let $a \in R$.

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

Then $aR$ is an right ideal of $R$.

Proof
We establish that $aR$ is an right ideal of $R$, by verifying the conditions of Test for Right Ideal.

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

Let $x, y \in ar$.

Then:

Let $s \in aR, x \in R$.

Thus by Test for Right Ideal, $aR$ is a right ideal of $R$.