I am asked to show the following:
$ab$ is a non-zero divisor of $R$ if and only if $a$ and $b$ are both non-zero divisors of $R$.
$\Rightarrow)$ Suppose $ab$ is a non-zero divisor of R. Then neither a nor b equal zero, thus $ab \neq 0.$ Also
$\exists c \in R$, $c \neq 0$ such that $(ab)c \neq 0.$ If $a(bc) \neq 0,$ then $bc \neq 0.$ Since $c \neq 0,$ we conclude that $b$ is a
non-zero divisor. Similarly, if $c(ab) \neq 0,$ then $(ca)b \neq 0.$ This implies that $ca \neq 0.$ Hence $a$ must
be a non-zero divisor.
$\Leftarrow )$ Suppose $a$ and $b$ are both non-zero divisors. Then $\exists d,e\in R, d \neq 0$ and $e\neq 0$ such that $ad \neq 0$
and $be \neq 0.$
I'm stuck on the second portion of the proof. Also, is the first half correct? Any clues on what to do next would be greatly appreciated. Thank you for your time.
Best Answer
Hint: For the second portion, try a proof by contradiction: For non-zero divisors $a, b \in R$, suppose $(ab)c=0$ for nonzero $c$, then by associativity $a(bc)=0$...