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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?

Student
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    Sometimes $0$ is considered a zero divisor, and sometimes it is not. – Elchanan Solomon Mar 02 '13 at 19:54
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    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 02 '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 02 '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 02 '13 at 20:05
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    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 02 '13 at 20:08
  • @Georges Said results on zero-divisors have been occasionally discussed here, but I don't recall any mention of said dire consequences. What did you have in mind? – Math Gems Mar 02 '13 at 20:24
  • Dear @Math Gems, thanks for the links. My mention of dire consequences is just a childish joke, in which I pretend to threaten the "heretics" who do not obey the orders of my gurus. – Georges Elencwajg Mar 02 '13 at 20:32
  • @Georges I thought you might know some hmorous examples. If Hardy is allowed to blowup $\Bbb Z$ with his infamous Theorem 90, we ring theorists should be permitted to top that by annihilating some rings with fuzzy zero-divisors! This worries even the greatest algebraists, e.g. Kaplanky wrote: "It is perhaps treacherous to try to talk about zero-divisors on the zero module, so (except no doubt for occasional forgetfulness) we shall not do so" (p.34, Commutative Rings) – Math Gems Mar 02 '13 at 21:01
  • Dear @Math Gems, thanks for the new link: I didn't know about Hardy's Theorem 90. Although I think consideration of the zero ring is vital, I concede that deciding whether $0$ is a zero-divisor in it is less clear than in other rings and Kaplansky's decision to avoid an analogous issue is certainly a wise one... – Georges Elencwajg Mar 02 '13 at 21:30

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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.

Thomas
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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?

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

gnometorule
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user2041143
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    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 02 '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 02 '13 at 23:59