Ring by Idempotent

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

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

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

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

By definition of generated ideal, $I$ is the intersection of all ideals of $A$ containing $e$.

Thus $e \mathop \in I$.

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

Then

It follows that $e$ is a unity of $\struct {I, +, \circ}$.

Hence $\struct {I, +, \circ}$ is a commutative ring with unity with unity $e$.