Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

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?

share|improve this question
Sometimes $0$ is considered a zero divisor, and sometimes it is not. –  Isaac Solomon Mar 2 '13 at 19:54
When is $0$ not considered a zero-divisor? I suppose sometimes you'll want to reference a non-trivial zero divisor, but it seems like always want $0$ to be considered a zero-divisor. @IsaacSolomon –  Thomas Andrews Mar 2 '13 at 19:57
Essentially, a number is "not a zero divisor" if you can always cancel it from an equation in the ring. –  Thomas Andrews Mar 2 '13 at 20:00
Thomas is spot on: zero definitely is a zero divisor according to any reasonable definition of the term. Moreover, many well-known theorems would be false if one took the absurd position that zero is not a zero-divisor. For example the result that in a noetherian ring the zero divisors consist of the union of the minimal primes. Need I point out the dire consequences for the heretics believing that Bourbaki, Atiyah-Macdonald, Matsumura,... are wrong on this ? –  Georges Elencwajg Mar 2 '13 at 20:05
To answer Student's question, the element $a\in R$ is not a zero-divisor iff the multiplication map $R\to R:x\mapsto ax$ is injective. –  Georges Elencwajg Mar 2 '13 at 20:08

3 Answers 3

up vote 3 down vote accepted

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.

share|improve this answer
Awesome thank you! –  Student Mar 2 '13 at 19:58
@Student: Glad to help –  Thomas Mar 2 '13 at 19:59

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?

share|improve this answer
Since $a\neq 0$, in order for $a(b-c) = 0$, it must be the case that $b-c = 0$ right? –  Student Mar 2 '13 at 19:52
Because $a$ is not a zero divisor, not because $a\neq 0$. $0$ is the prototypical zero-divisor. @Student –  Thomas Andrews Mar 2 '13 at 19:53
@ThomasAndrews: I see now. Thanks! –  Student Mar 2 '13 at 19:55

$b-c=0$ because any number - that number gives $0$. Else you can't get $0$ if $b>c$ or $c>b$.

share|improve this answer
While it is true that $x-x=0$, this is not what was asked here. Also, note that we are talking about general rings and not numbers: The notion $b>c$ might not even make sense in the given ring. For example, in $\mathbb Z/2\mathbb Z$, there is no ordering compatible with addition. –  Johannes Kloos Mar 2 '13 at 19:53
Oh.. I didn't understood his question then, as for the >, I made a mistake, I should have said if a is different than b. –  user2041143 Mar 2 '13 at 23:59

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.