# Product is Zero Divisor means Zero Divisor

 $\displaystyle \paren {x \circ y}$ $\divides$ $\displaystyle 0_R$ $\displaystyle \leadsto \ \$ $\displaystyle \exists z \divides 0_R \in R: \paren {x \circ y} \circ z$ $=$ $\displaystyle 0_R, x \ne 0_R, y \ne 0_R$ Definition of Zero Divisor of Ring $\displaystyle \leadsto \ \$ $\displaystyle x \circ \paren {y \circ z}$ $=$ $\displaystyle 0_R$ Ring Axiom $M1$: Associativity of $\circ$ $\displaystyle \leadsto \ \$ $\displaystyle x \divides 0_R$ $\lor$ $\displaystyle \paren {y \circ z} \divides 0_R$ Definition of Zero Divisor of Ring $\displaystyle \leadsto \ \$ $\displaystyle x \divides 0_R$ $\lor$ $\displaystyle y \divides 0_R$ Zero Product with Proper Zero Divisor is with Zero Divisor applies, as $z \divides 0_R$
$\blacksquare$