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.


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