Zero Divisor Product is Zero Divisor

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Theorem

The ring product of a zero divisor with any ring element is a zero divisor.


Proof

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

Let $x \divides 0_R$ in $R$. Then:

\(\, \displaystyle \exists y \in R, y \ne 0: \, \) \(\displaystyle x \circ y\) \(=\) \(\displaystyle 0_R\) Definition of Zero Divisor of Ring
\(\displaystyle \leadsto \ \ \) \(\, \displaystyle \forall z \in R: \, \) \(\displaystyle z \circ \paren {x \circ y}\) \(=\) \(\displaystyle z \circ 0_R\)
\(\displaystyle \) \(=\) \(\displaystyle 0_R\) Property of Zero
\(\displaystyle \leadsto \ \ \) \(\, \displaystyle \forall z \in R: \, \) \(\displaystyle \paren {z \circ x} \circ y\) \(=\) \(\displaystyle 0_R\) Associativity of $\circ$

So $z \circ x \divides 0_R$ in $R$.

The same thing happens if we form the product $\paren {x \circ y} \circ z$.

$\blacksquare$