Most definitions of characteristic I come across are along the following lines:

A ring $R$ has characteristic $n\geqslant 1$ if $n$ is the least positive integer satisfying $$nx=0$$ for all $x\in R$, and that $R$ has characteristic 0 otherwise.

Now, the definition I recall from my undergraduate study is different: we said that $R$ has characteristic 0 if each non-zero element $x\in R$ satisfies $nx\neq 0$ for all $n\in\mathbb N$. In other words, for us, characteristic 0 is equivalent to torsion-free; so we would say that, e.g., $R=(\mathbb Z/2\mathbb Z)\times \mathbb Z$ simply does not have a characteristic, since $n(0,1)$ is never $0$, yet $2(1,0)=0$.

Notice that the "usual" (Wikipedia) definition and my definition are in agreement when $R$ is an integral domain, the only time this makes a difference is when $R$ is not an integral domain.

The definition we did is in accordance with that given in the textbook Topics in Algebra by Herstein, who says that the concept of characteristic for arbitrary rings is meaningless:

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Now, some people argue that the definition I give (which implicitly refrains from assigning a characteristic to rings with torsion elements) is an incorrect one, see for instance this answer.

So, my question is: why is it useful to say that such rings have characteristic 0, as opposed to leaving the characteristic undefined?

Luke Collins
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  • See also https://math.stackexchange.com/questions/98605/why-characteristic-zero-and-not-infinite-characteristic – lhf Apr 29 '22 at 13:42
  • Undefined is not our friend in mathematics. – Wuestenfux Apr 29 '22 at 13:48
  • @lhf That question is quite different to mine, I'm asking why does it make sense to consider the notion of characteristic 0 for rings which are not integral domains. – Luke Collins Apr 29 '22 at 13:48
  • It probably doesn't matter all that much. The notion of characteristic is most relevant for distinguishing between very differently behaved fields (esp. the characteristic $0$ versus characteristic $p>0$ distinction). Non-field examples where characteristic is relevant for classifying results will likely be integral domains, in the situation of mixed characteristic: https://en.wikipedia.org/wiki/Ring_of_mixed_characteristic . – Karl Kroningfeld Apr 30 '22 at 00:55

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