Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

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?
share|cite|improve this question
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.

share|cite|improve this answer
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

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.