# 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:

 $\, \displaystyle \exists y \in R, y \ne 0: \,$ $\displaystyle x \circ y$ $=$ $\displaystyle 0_R$ Definition of Zero Divisor of Ring $\displaystyle \leadsto \ \$ $\, \displaystyle \forall z \in R: \,$ $\displaystyle z \circ \paren {x \circ y}$ $=$ $\displaystyle z \circ 0_R$ $\displaystyle$ $=$ $\displaystyle 0_R$ Property of Zero $\displaystyle \leadsto \ \$ $\, \displaystyle \forall z \in R: \,$ $\displaystyle \paren {z \circ x} \circ y$ $=$ $\displaystyle 0_R$ Associativity of $\circ$

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

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

$\blacksquare$