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

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

$\blacksquare$