Zero Divisor Product is Zero Divisor

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

Proof
Let $\struct {R, +, \circ}$ be a ring.

Let $x \divides 0_R$ in $R$. Then:

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

The same thing happens if we form the product $\paren {x \circ y} \circ z$.