Zero Product with Proper Zero Divisor is with Zero Divisor

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Let $\struct {R, +, \circ}$ be a ring.

Let $x \in R$ be a proper zero divisor of $R$.


$\paren {x \divides 0_R} \land \paren {x \circ y = 0_R} \land \paren {y \ne 0_R} \implies y \divides 0_R$

That is, if $x$ is a proper zero divisor, then whatever non-zero element you form the product with it by to get zero must itself be a zero divisor.


Follows directly from the definition of proper zero divisor.

If $y \ne 0_R$ and $x \circ y = 0_R$ and $x \in R^*$ (which is has to be if it's a proper zero divisor), then all the criteria of being a zero divisor are fulfilled by $y$.