Editing Characteristic (algebra) (section)

== Case of rings ==
If {{math|''R''}} and {{math|''S''}} are [[ring (mathematics)|rings]] and there exists a [[ring homomorphism]] {{math|''R'' → ''S''}}, then the characteristic of {{math|''S''}} divides the characteristic of {{math|''R''}}. This can sometimes be used to exclude the possibility of certain ring homomorphisms. The only ring with characteristic {{math|1}} is the [[zero ring]], which has only a single element {{math|0}}. If a nontrivial ring {{math|''R''}} does not have any nontrivial [[zero divisor]]s, then its characteristic is either {{math|0}} or [[prime number|prime]]. In particular, this applies to all [[field (mathematics)|fields]], to all [[integral domain]]s, and to all [[division ring]]s. Any ring of characteristic zero is infinite.

The ring <math>\mathbb{Z}/n\mathbb{Z}</math> of integers [[modular arithmetic|modulo]] {{math|''n''}} has characteristic {{math|''n''}}. If {{math|''R''}} is a [[subring]] of {{math|''S''}}, then {{math|''R''}} and {{math|''S''}} have the same characteristic. For example, if {{math|''p''}} is prime and {{math|''q''(''X'')}} is an [[irreducible polynomial]] with coefficients in the field <math>\mathbb F_p</math> with {{mvar|p}} elements, then the [[quotient ring]] <math>\mathbb F_p[X]/(q(X))</math> is a field of characteristic {{math|''p''}}. Another example: The field <math>\mathbb{C}</math> of [[complex number]]s contains <math>\mathbb{Z}</math>, so the characteristic of <math>\mathbb{C}</math> is {{math|0}}.

A <math>\mathbb{Z}/n\mathbb{Z}</math>-algebra is equivalently a ring whose characteristic divides {{math|''n''}}. This is because for every ring {{math|''R''}} there is a ring homomorphism <math>\mathbb{Z}\to R</math>, and this map factors through <math>\mathbb{Z}/n\mathbb{Z}</math> if and only if the characteristic of {{math|''R''}} divides {{math|''n''}}. In this case for any {{math|''r''}} in the ring, then adding {{math|''r''}} to itself {{math|''n''}} times gives {{math|''nr'' {{=}} 0}}.

If a commutative ring {{math|''R''}} has ''prime characteristic'' {{math|''p''}}, then we have {{math|(''x'' + ''y''){{i sup|''p''}} {{=}} ''x''{{i sup|''p''}} + ''y''{{i sup|''p''}}}} for all elements {{math|''x''}} and {{math|''y''}} in {{math|''R''}} – the normally incorrect "[[freshman's dream]]" holds for power {{math|''p''}}.
The map {{math|''x'' ↦ ''x''{{i sup|''p''}}}} then defines a [[ring homomorphism]] {{math|''R'' → ''R''}}, which is called the ''[[Frobenius homomorphism]]''. If {{math|''R''}} is an [[integral domain]] it is [[injective]].