Product is Zero Divisor means Zero Divisor

Theorem

If the ring product of two elements of a ring is a zero divisor, then one of the two elements must be a zero divisor.

Proof

	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \left({x \circ y}\right)\)	\(\backslash\)	\(\displaystyle 0_R\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)
	\(\displaystyle \)	\(\displaystyle \implies\)	\(\displaystyle \)	\(\displaystyle \exists z \backslash 0_R \in R: \left({x \circ y}\right) \circ z\)	\(=\)	\(\displaystyle 0_R, x \ne 0_R, y \ne 0_R\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	Definition of Zero Divisor
	\(\displaystyle \)	\(\displaystyle \implies\)	\(\displaystyle \)	\(\displaystyle x \circ \left({y \circ z}\right)\)	\(=\)	\(\displaystyle 0_R\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	Associativity of $\circ$
	\(\displaystyle \)	\(\displaystyle \implies\)	\(\displaystyle \)	\(\displaystyle x \backslash 0_R\)	\(\lor\)	\(\displaystyle \left({y \circ z}\right) \backslash 0_R\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	Definition of Zero Divisor
	\(\displaystyle \)	\(\displaystyle \implies\)	\(\displaystyle \)	\(\displaystyle x \backslash 0_R\)	\(\lor\)	\(\displaystyle y \backslash 0_R\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	Zero Divisor Product applies, as $z \backslash 0_R$

$\blacksquare$