# Pairs of 0 Divisors in a Ring

• pivoxa15
As for your lecturer, I would guess that he was trying to say that 0=1 doesn't happen very often, which is true, but I would personally just say that 0 and 1 are different unless the ring is the trivial ring.

#### pivoxa15

Definiton
If a and b are two nonzero elements of a ring R such that ab=0 then a and b are divisors of 0 or (0 divisors). In particular a is a left divisor of 0 and b is a right divisor of 0.

My question is if you have a0=0 then a is not a 0 divisor?

So it seems that 0 divisors will always come in pairs? So whenever you have a left 0 divisor, you automatically have a right 0 divisor.

As for your question, if a*0=0, a may or may not be a zero divisor.
Eg if a = 0, then a*0 = 0*0=0 and a is not a zero divisor.
Eg if a is a left zero divisor, than a*0=0 still.

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I realize my question in my OP isn't precise enough. I will set up my question better.

Consider ac=0 for a is not 0 and c is arbitary. a,c part of ring R.

If R satisfies the cancellation law then
ac=0
ac=a0, since a0=0
c=0

So c is forced to be 0 and so a is not a zero divisor hence no left 0 divisors in R.

If there are no left 0 divisors in R then there cannot be right 0 divisors in R. -Correct?
In this way 0 divisors in rings must come in pairs. i.e. in any ring there must be an even number of 0 divisors?

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All domains have exactly one zero divisor. :tongue:

There are examples of either parity: consider Z mod 4 (0 and 2) and Z mod 9. (0, 3, 6)

pivoxa15 said:
If there are no left 0 divisors in R then there cannot be right 0 divisors in R. -Correct?

Yes. Just look at the definition.

In this way 0 divisors in rings must come in pairs. i.e. in any ring there must be an even number of 0 divisors?

of course not. try to think why.

Hurkyl said:
All domains have exactly one zero divisor. :tongue:

What does this mean? Could you be more specifc? Or is it meant to be a joke, in which case I don't understand.
Hurkyl said:
There are examples of either parity: consider Z mod 4 (0 and 2) and Z mod 9. (0, 3, 6)

Are you claiming 0 is a 0 divisor? The definition clearly states that 0 divisors must not be 0. i.e. ab=0 where a,b are non zero. => a,b are 0 divisors.

Although with Z mod 4, I see that 2 is the only zero divisor. i.e 2*2=0.

However, it still is the case that whenever you have a left 0 divisor, you must also have a right 0 divisor. In this example, the left and right 0 divisors turned out to be the same.

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Your definition of zero divisor above is not one I would ever use. x is a left zero divisor if xy=0 for some non-zero y. Similarly for a right zero divisor. Excluding 0 as a zero divisor seems very odd to me. I suspect the book chose it to clean up the presentation of statements (i.e. 'let x be a (left) zero divisor' instead of 'let x be a non-trivial (left) zero divisor').

matt grime said:
Your definition of zero divisor above is not one I would ever use. x is a left zero divisor if xy=0 for some non-zero y. Similarly for a right zero divisor. Excluding 0 as a zero divisor seems very odd to me. I suspect the book chose it to clean up the presentation of statements (i.e. 'let x be a (left) zero divisor' instead of 'let x be a non-trivial (left) zero divisor').

If you let 0 as a 0 divisor then you run into 0/0 (as 0 is a 0 divisor) which is undefined. So you shouldn't include 0 as a 0 divisor.

Erm, no, that is nonsense. At no point do you run into 0/0 issues anymore than any other issues. Let me explain:

Just think mod 12. 4*3=0 and 4*9=0 mod 12. So you can't divide by 4 (if you could then 3=9 mod 12). This is the whole point: you can't divide by zero divisors! Zero divisors are a generalization of the fact you can't divide by 0.

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I see. You can't divide by 3 nor 9 because you will get 4=0 Mod 12 in both cases.

So if you include 0 as a zero divisor than it is a trivial zero divisor. In fact 0 is always a 0 divisor in whatever ring you may have. That is why Hurkyl said "All domains have exactly one zero divisor." This zero divisor is 0.

However in here http://en.wikipedia.org/wiki/Zero_divisor they used the definition I gave in my OP which is that zero divisors must be nonzero. Are you going to edit this entry Matt?

I agree with matt -- it's likely just another convention. Like whether people let 0 be a natural number, or if 0=1 is allowed in a ring. (and I greatly prefer the 0 is a zero divisor version)

My lecturer specifically said that 0=1 in a ring R if and only if R={0}. This is because 0 in this ring can act as the multiplication identiy associated with 1.

On p58 of Intro algebra by Hungerford it states the definition I gave and with an added note "Note that 0 is not a zero divisor". So Hungerford is against the notion of 0 as a zero divisor. Maybe it's because of the trivialness of 0 being a 0 divisor.

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No, it's probably because if makes it easier to state things, just like we exclude units from being primes, even though they satisfy the definition of prim (p is prime if p divides ab implies p divides a or b - units divide everything). Since 0 is always a zero divisor, and you will be specifically interested in finding non-zero zero divisors, it is better to exclude them for presentational reasons. I've never had a situation where I've seen them exclude zero before.

## 1. What is a ring?

A ring is an algebraic structure consisting of a set of elements and two binary operations, usually called addition and multiplication, that satisfy certain properties. Rings are used in abstract algebra to study properties and patterns of numbers and other mathematical objects.

## 2. What are 0 divisors in a ring?

0 divisors in a ring are elements that, when multiplied together, result in 0. This means that the product of two 0 divisors is always 0, and they cannot have a multiplicative inverse.

## 3. Can a ring have more than one pair of 0 divisors?

Yes, a ring can have multiple pairs of 0 divisors. This occurs when there are multiple elements that, when multiplied together, result in 0.

## 4. How do pairs of 0 divisors affect the properties of a ring?

Pairs of 0 divisors can affect the properties of a ring in various ways. For example, if a ring has a pair of 0 divisors, it cannot be an integral domain, as there exist non-zero elements that multiply to 0. Additionally, the presence of 0 divisors can affect the existence and uniqueness of solutions to equations in the ring.

## 5. Can a ring without 0 divisors still have a 0 element?

Yes, a ring without 0 divisors can still have a 0 element. This means that there exists an element that, when multiplied by any other element in the ring, results in 0. However, in this case, the 0 element is not considered a 0 divisor, as it does not have a matching element that multiplies to 0.