# Categorical definition of the characteristic of a ring

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?
-

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.
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