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I was solving the following problem given at IMC $2000$ :

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Why the definition of characteristic zero is different from the one given on Wikipedia ? It is clear that the $2$ definitions are not equivalent:

Let us consider the ring $A = Z[X]/2X$ ; using the definition given in the problem , $A$ doesn't have characteristic $0$ because $2X=0$, but using the other one we obtain that $A$ has characteristic $0$

waffle235
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1 Answers1

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The definition from "problem 5" is indeed highly problematic. In particular, it suggests that a ring of characteristic $0$ cannot have any elements with finite additive order, but that is not accurate.

The one in Wikipedia is right: it should be the minimum bound on the additive orders of all elements in the ring, if it exists, and $0$ otherwise.

For example, $\mathbb Z\times F_2$ has $x=(0,1)$ which satisfies $2x=0$, but the characteristic of this ring (using the 'right' definition) is $0$.

The definition using "minimal bound on additive orders" works even in rings without identity, but it is worth knowing that in rings with identity, you can immediately spot the characteristic by checking the additive order of the identity. I'll leave this equivalence up to you.


With the normal definition of characteristic, the cited problem is incorrect.

If $R=F_2\times F_2\times \mathbb Z$, $R$ has characteristic $0$, but $e=(1,0,0)$, $f=(0,1,0)$, and $g=(1,1,0)$ provide a counterexample to the claim.

Perhaps they intended to say "no element except zero has finite additive order." This is what is called a torsion-free ring.

rschwieb
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    An approach.xyz search immediately turned up an attempted solution [here](https://artofproblemsolving.com/community/c7h58510p357641) that tries to use the characteristic to conclude 3 is invertible... but it isn't necessarily. The given definition of characteristic is strong enough to make that argument go through, though. – rschwieb May 06 '20 at 14:04