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?
Best Answer
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:
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.