Are 0 divisors always paired in a ring?

[pivoxa15]

definition
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.

 

[gammamcc]

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.

 

[pivoxa15]

Your second example is interesting.

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?

 

[Hurkyl]

All domains have exactly one zero divisor.

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

 

[matt grime]

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.

 

[pivoxa15]

All domains have exactly one zero divisor.

What does this mean? Could you be more specifc? Or is it meant to be a joke, in which case I don't understand.

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.

 

[matt grime]

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').

 

[pivoxa15]

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.

 

[matt grime]

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.

 

[pivoxa15]

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?

 

[Hurkyl]

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)

 

[pivoxa15]

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.

 

[matt grime]

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.