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The characteristic of a ring is an important algebraic concept (and a specific number), but it refers to elements, so - in my understanding - it is evil (from the point of view of category theory). So the question arises:

How is the characteristic of a ring defined and calculated categorically?

Given the category of rings, how is the characteristic - as a number - of its objects calculated:

  • from their in- and outgoing arrows?
  • from special functors?
  • from or to which other categories?
  • or what ever?
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1 Answer 1

up vote 5 down vote accepted

A ring $R$ has characteristic 0 iff the morphism from the initial object $Z$ of the category of rings is a monomorphism. In general the morphism $Z\to R$ factors (in unique way) as $Z\to P\to R$ where $Z\to P$ is a regular epi and $P\to R$ is mono. The characteristic of $R$ is just this morphism $Z\to P$, if you will.

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Or, better, the characteristic of $R$ is the kernel of the unique homomorphism $\mathbb{Z} \to R$. –  Zhen Lin Nov 6 '12 at 0:02
    
@ZhenLin Certainly -- but what do you mean by kernel in the category of rings? –  Grigory M Nov 6 '12 at 7:31
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