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

$$ $$ $$ $$

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

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