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Characteristic (algebra)
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== Motivation == The special definition of the characteristic zero is motivated by the equivalent definitions characterized in the next section, where the characteristic zero is not required to be considered separately. The characteristic may also be taken to be the [[exponent (group theory)|exponent]] of the ring's [[additive group]], that is, the smallest positive integer {{math|''n''}} such that:<ref name=Fraleigh-Brand-2020> {{cite book |first1=John B. |last1=Fraleigh |first2=Neal E. |last2=Brand |year=2020 |title=A First Course in Abstract Algebra |edition=8th |publisher=[[Pearson Education]] |url=https://www.pearson.com/us/higher-education/program/Fraleigh-Pearson-e-Text-First-Course-in-Abstract-Algebra-A-Access-Card-8th-Edition/PGM282304.html }} </ref>{{rp|style=ama|p=β―198, Def.β―23.12}} : <math>\underbrace{a+\cdots+a}_{n \text{ summands}} = 0</math> for every element {{math|''a''}} of the ring (again, if {{math|''n''}} exists; otherwise zero). This definition applies in the more general class of [[Rng (algebra)|rng]]s (see ''{{slink|Ring (mathematics)#Multiplicative identity and the term "ring"}}''); for (unital) rings the two definitions are equivalent due to their [[distributive law]].
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