Set of Ring Elements forming Zero Product with given Element is Ideal

Theorem
Let $\struct {R, +, \circ}$ be a commutative ring whose zero is $0_R$.

Let $a \in R$ be an arbitrary element of $R$.

Let $A$ be the subset of $R$ defined as:


 * $A = \set {x \in R: x \circ a = 0_R}$

Then $A$ is an ideal of $A$.

Proof
By definition of ring zero:
 * $\forall x \in R: x \circ 0_R = 0_R$

Hence $0_R \in A$ and so $A \ne \O$.

Let $a, b \in A$.

Thus:

Then:

Hence the result, from Test for Ideal: