Zero Divisor Product is Zero Divisor

Theorem
The ring product of a zero divisor with any ring element is a zero divisor.

Proof
Let $\left({R, +, \circ}\right)$ be a ring.

Let $x \mathop \backslash 0_R$ in $R$. Then:

So $z \circ x \mathop \backslash 0_R$ in $R$.

The same thing happens if we form the product $\left({x \circ y}\right) \circ z$.