Characteristic (algebra) - Other Equivalent Characterizations

Other Equivalent Characterizations

• The characteristic is the natural number n such that nZ is the kernel of a ring homomorphism from Z to R;
• The characteristic is the natural number n such that R contains a subring isomorphic to the factor ring Z/nZ, which would be the image of that homomorphism.
• When the non-negative integers {0, 1, 2, 3, . . . } are partially ordered by divisibility, then 1 is the smallest and 0 is the largest. Then the characteristic of a ring is the smallest value of n for which n · 1 = 0. If nothing "smaller" (in this ordering) than 0 will suffice, then the characteristic is 0. This is the right partial ordering because of such facts as that char A × B is the least common multiple of char A and char B, and that no ring homomorphism ƒ : A → B exists unless char B divides char A.
• The characteristic of a ring R is n ∈ {0, 1, 2, 3, . . . } precisely if the statement ka = 0 for all a ∈ R implies n is a divisor of k.

The requirements of ring homomorphisms are such that there can be only one homomorphism from the ring of integers to any ring; in the language of category theory, Z is an initial object of the category of rings. Again this follows the convention that a ring has a multiplicative identity element (which is preserved by ring homomorphisms).