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 $z \divides 0_R$ |
$\blacksquare$