Ring by Idempotent

Theorem
Let $\struct {A, +, \circ}$ be a commutative ring.

Let $e$ be an idempotent element of $A$.

Then the ideal $e A := \ideal e$ generated by $e$ is a commutative ring with unity $\struct {e A, +, \circ}$ with unit $e$.

Proof
Because $\struct {e A, +}$ is an ideal of $\struct {e A, +, \circ}$, it follows that $\struct {e A, +, \circ}$ is a ring (not necessarily unital).

Clearly $e \in eA$.

Let $e a \in e A$ for some arbitrary $a \in A$.

Then

It follows that $e$ is a unit of $e A$.

Hence commutative ring with unity $e$.