TITLE: Not a Zero Divisor
QUESTION [6 upvotes]: Let $R$ be a commutative ring. Then we say $a \in R$ is a zero divisor if there exists $b \neq 0$ such that $ab = 0$.
I want to know what it means to not be a zero divisor. So I tried  to negate the statement: $a$ is not a zero divisor if for every $b \neq 0$ we have $ab \neq 0$.
Also taking the contrapositive of the initial statement I got the following: If for every $b \neq 0$, $ab \neq 0$, then $a$ is not a zero divisor.
Have I negated the definition of a zero divisor and taken the contrapositive correctly?

My book has the following theorem: Suppose $a$ is not a zero-divisor. Then if $ab = ac$, we can conclude that $b = c$. 
Proof: $ab - ac = a(b-c) = 0$. Since $a$ is not a zero-divisor, $b-c = 0$ so $b=c$. 
I don't see why $b-c = 0$ because $a$ is not a zero-divisor. Could someone explain?

REPLY [4 votes]: Yes a zero divisor is an element $a\neq 0$ such that you can find a $b\neq 0$ with $ab\ = 0$. The existence of zero divisors in a ring just means that the product of two non-zero elements can be zero.
So indeed, as you write, $a\neq 0$ is not a zero divisor if one of the following equivalent statements are satisfied:

There does not exist a $b\neq 0$ such that $ab = 0$.
$ab = 0$ implies that $b = 0$.
$b\neq 0$ implies $ab \neq 0$.

So indeed is given $a\neq 0$ satisfies that all $b\neq 0$ you have that $ab\neq0$ then $a$ is not a zero divisor.

REPLY [2 votes]: Yes, you have determined the correct formulation for what it means to be a non-zero-divisor.
If $a$ is not a zero-divisor, then for every $r\neq 0$, we have that $ar\neq 0$. But $ar=0$ when $r=b-c$. What does that tell you?