Principal Left Ideal is Left Ideal

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

Let $a \in R$.

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

Then $Ra$ is an left ideal of $R$.

Proof
We establish that $Ra$ is an left ideal of $R$, by verifying the conditions of Test for Left Ideal.

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

Let $x, y \in Ra$.

Then:

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

Thus by Test for Left Ideal, $Ra$ is a left ideal of $R$.

Also see

 * Principal Right Ideal is Right Ideal where it is shown that a principal right ideal is a right ideal