Characteristic of a ring
Let RING be the category of unital rings, and an object in it. The object is initial in RING: for every ring there is exactly one momorphism . Thus we can define the characteristic of to be the unique non-negative generator of the ideal in the PID .
In view of the induced isomorphism , we see that if has no zero-divisors then neither does , forcing the latter to be a domain, so must be prime or zero.
In particular, let be a field. The embedding extends (by the universal property of quotient field) to a field embedding of the quotient field .
We see that contains a subfield isomorphic to if is prime, and to if ; called the prime field of . A subfield has the same characteristic since has the same kernel as , therefore the prime field of is characterized as its smallest subfield.
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- May 18, 2010 / 9:48 pm