Ring theory MOC

Characteristic

The characteristic of a rng is the smallest positive integer such that the sum of copies of any is , i.e. . #m/def/ring If no such exists then . For a ring with unity, the characteristic is the additive group order of unity (or zero if the order is infinite).

Proof

If has infinite additive order, then there is no such that and thus . Now suppose that has additive order , i.e. is the smallest positive integer such that and thus . Now for any

hence .

Properties

  1. The characteristic of an integral domain is 0 or prime
  2. ^P1 (this gives a nice alternative definition of characteristic for a ring)

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