Why is $0$ a zero divisor?

Question

Why do some sources define zero divisors in such a way that $0$ is a zero divisor? Does it not make things confusing? For example, in this terminology integral domains actually have zero divisors, but have no non-zero zero divisors. To me it seems that since $0$ acts "the same" in any ring, then there is nothing to win from bringing it "into equation".

Answer 1

Caution, the subring {0} of ℤ has no zero divisor. ℤ₁ has no zero divisor, too.

The essential notion behind all 'zero divisor' terms seems to be this, R ring: It's trivial that each element x of R left-divides 0 in the sense that 0=xy for y=0, but it's much less trivial that there's an element y≠0. (The same holds for right division.)

I think of 'ring zero divisor' as any 'element capable of multiplicatively reaching 0 without resorting to 0 as multiplier'.

Edit: There's been some debate about proper definition of 'zero divisor'. Philosophically, this should be taken seriously. After all, maybe we are dealing with (slightly) different concepts? Above, I present one such concept.

Answer 2

I think the whole issue of whether $0$ is or is not a $0$-divisor can be resolved by just a few added words of context... rather than trying to achieve some perfect, absolute definition. (Similar to how $1$ was conventionally a prime until some point in the 19th century, when people collectively agreed that it was more convenient, for statement of various theorems, to say that $1$ is not prime.

Similarly, I'd advocate saying $0$ is not a proper $0$-divisor, although, yes, by anyone's definition of divisibility, $0$ does divide $0$. That's just not interesting. :)

In recent years, I've more-and-more tried to emphasize to students that there is an enormous difference between conventions ("definitions"? terminology?) and mathematical facts. Facts do not depend on context/language. :)

E.g., $R=\{0\}$ is a ring, by some definitions, but $0=1$. Maybe it's a field, too? But/and if we declare that "field" only allows $0\not=1$, then what is this thing? :)

And $3\mathbb Z/6$ is certainly some kind of subring of $\mathbb Z/6$, and it has unit $3$ (mod $6$), but that unit is not the unit of $\mathbb Z/6$. Nothing pathological here... but only illustrating the mathematical fact that ring (with unit) inclusions need not map $1$ to $1$, so that that might have to be added as a requirement, if it is desired. :)