Product is Zero Divisor means Zero Divisor

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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

\(\displaystyle \paren {x \circ y}\) \(\divides\) \(\displaystyle 0_R\)
\(\displaystyle \leadsto \ \ \) \(\displaystyle \exists z \divides 0_R \in R: \paren {x \circ y} \circ z\) \(=\) \(\displaystyle 0_R, x \ne 0_R, y \ne 0_R\) 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 $z \divides 0_R$

$\blacksquare$