# Characteristic of a ring

Let RING be the category of unital rings, and $R$  an object in it. The object $\mathbb{Z}$ is initial in RING: for every ring $S$ there is exactly one momorphism $f_S:\mathbb{Z}\to S$. Thus we can define the characteristic $c_R$ of $R$  to be the unique non-negative generator of the ideal $\ker f_R$ in the PID $\mathbb{Z}$.

In view of the induced isomorphism $\mathbb{Z}/c_R\mathbb{Z}\xrightarrow{\cong}f_R(\mathbb{Z})\subset R$, we see that if $R$ has no zero-divisors then  neither does $\mathbb{Z}/c_R\mathbb{Z}$, forcing the latter to be a domain, so $c_R$ must be prime or zero.

In particular, let  $R=k$ be a field. The embedding  $f_k:\mathbb{Z}/c_k\mathbb{Z}\hookrightarrow k$ extends (by the universal property of quotient field) to a field embedding of the quotient field $\mathcal{Q}(\mathbb{Z}/c_k\mathbb{Z})$.

We see that $k$ contains a subfield isomorphic to $\mathbb{Z}/p\mathbb{Z}$ if $c_k=p$ is prime, and to $\mathbb{Q}$ if $c_k=0$; called the prime field of $k$. A subfield $k'\subset k$ has the same characteristic since $f_{k'}:\mathbb{Z}\to k'\to k$ has the same kernel as $f_k$, therefore the prime field of $k$ is characterized as its smallest subfield.