Editing Characteristic (algebra) (section)

== Equivalent characterizations ==
* The characteristic of a ring {{math|''R''}} is the [[natural number]] {{math|''n''}} such that {{math|''n''<math>\mathbb{Z}</math>}} is the [[kernel (ring theory)|kernel]] of the unique [[ring homomorphism]] from <math>\mathbb{Z}</math> to {{math|''R''}}.{{efn|The requirements of ring homomorphisms are such that there can be only one (in fact, exactly one) homomorphism from the ring of integers to any ring; in the language of [[category theory]], <math>\mathbb{Z}</math> is an [[initial object]] of the [[category of rings]]. Again this applies when a ring has a multiplicative identity element (which is preserved by ring homomorphisms).}}
* The characteristic is the [[natural number]] {{math|''n''}} such that {{math|''R''}} contains a [[subring]] [[ring homomorphism|isomorphic]] to the [[factor ring]] <math>\mathbb{Z}/n\mathbb{Z}</math>, which is the [[image (mathematics)|image]] of the above homomorphism.
* When the non-negative integers {{math|{{mset|0, 1, 2, 3, ...}}}} are [[Partially ordered set|partially ordered]] by divisibility, then {{math|1}} is the smallest and {{math|0}} is the largest.  Then the characteristic of a ring is the smallest value of {{math|''n''}} for which {{math|''n'' &sdot; 1 {{=}} 0}}. If nothing "smaller" (in this ordering) than {{math|0}} will suffice, then the characteristic is&nbsp;{{math|0}}. This is the appropriate partial ordering because of such facts as that {{math|char(''A'' × ''B'')}} is the [[least common multiple]] of {{math|char ''A''}} and {{math|char ''B''}}, and that no ring homomorphism {{math|''f'' : ''A'' → ''B''}} exists unless {{math|char ''B''}} divides {{math|char ''A''}}.
* The characteristic of a ring {{math|''R''}} is {{math|''n''}} precisely if the statement {{math|''ka'' {{=}} 0}} for all {{math|''a'' ∈ ''R''}} implies that {{math|''k''}} is a multiple of {{math|''n''}}.