Why is it called "characteristic $0$" and not "characteristic $\infty$?"

Question

There may not be a good answer to this; it is perhaps more or less a matter of coincidence. But the thing is: For any ring, the characteristic of the ring is the order of the element $1$ in the ring --- except if this order is $\infty$, in which case the characteristic is zero. So wouldn't it be more natural to call it characteristic $\infty$? Is there some smart reason behind this that I just haven't noticed yet, or is it just because mathematicians are weird?

I'm a mathematician myself, so I have the right to say it! – Gaussler Jan 10 '15 at 18:23 — Gaussler, Jan 10 '15 at 18:23

score 9 · Accepted Answer · answered Jan 10 '15 at 18:25

What is actually important is not the characteristic as a number, but the kernel of the unique ring homomorphism $\mathbb Z\to R$, as an ideal.

Since $\mathbb Z$ is a PID, this kernel is always a principal ideal, so we can express what it is by quoting its (nonnegative) generator. This generator is what we call the characteristic of the ring.

For a characteristic-0 ring, the kernel is $\{0\} = \langle 0\rangle$.

(In some sense this is the root cause of a number of cases where zero "behaves like" an infinity in ring theory).

That's a good reason indeed! – Gaussler Jan 10 '15 at 18:26 — Gaussler, Jan 10 '15 at 18:26

Why is it called "characteristic $0$" and not "characteristic $\infty$?"

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