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If for a ring $R$ a positive integer $n$ exists such that $n\cdot r = 0$ for all $r\in R$, then the least such positive integer is the characteristic of the ring $R$, denoted by ${\rm char}(R)$. If no such positive integer exists, then $R$ is of characteristic $0$.

I have seen repeatedly in papers that the authors assume that ${\rm char}(R)\neq 2$. I was wondering if somoene could explain me why they consider this hypthoethesis?

Mahtab
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    ...because they really want $2\ne 0$? – Trebor Mar 19 '24 at 11:58
  • This answer is one good reason - if you're in an integral domain and you want to conclude that $x = 0$ from $x = -x$, then you need this assumption. In fields, you need this assumption to make use of the polarisation identity, for example. – Izaak van Dongen Mar 19 '24 at 12:03
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    Here's a plan of action: take one of those papers where you've seen the authors assume the characteristic is not two, try to go through their arguments for the case when the characteristic is two, and see for yourself what goes wrong. Or, look at what they are trying to prove, and see whether you can come up with a counterexample if the characteristic is two. Then, report back to us! Post an answer to your own question! – Gerry Myerson Mar 19 '24 at 12:08
  • @IzaakvanDongen Thank you very much foe your nice comment and the links. I got it now. – Mahtab Mar 19 '24 at 12:12
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    @GerryMyerson You've got a point there. I'll do it. Appreciate your answer. – Mahtab Mar 19 '24 at 12:14
  • As another example from multilinear algebra: In a field of characteristic $\mathrm{char}(\mathbb{F})=2$, a bilinear form $\beta:V\times V\to\mathbb{F}$ on some $\mathbb{F}$-vector space $V$ is symmetric if and only it is antisymmetric, as you can check quite easily. Especially when studying objects like exterior algebras or Clifford algebras one hence often exlcludes this case from the discussion, because it behaves quite differently. – B.Hueber Mar 19 '24 at 12:25
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    @B.Hueber I understood well. Thank you so much for your nice comment. – Mahtab Mar 19 '24 at 15:33

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