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Definition: Characteristic of a Ring

Let $$(R, +,\cdot)$$ be a ring free of zero divisors and let $$1$$ be its multiplicative and $$0$$ its additive identity. Then the minimal1 natural number $$n$$, for which
$\underbrace{1 + \ldots + 1}_{n\text{ times}}=n\cdot 1=0$
is called the characteristic of $$R$$, and denoted by $\operatorname{char}( R ):=n.$

1 Such minimal element exists due to the well-ordering principle of the natural numbers. If in a given ring $$R$$ there is no such number with $$n > 0$$, then $$n=0$$ is the minimal number with this property. In this case we set $$char( R )=0$$, see also explanation why.

| | | | | created: 2014-09-17 21:45:32 | modified: 2019-08-10 08:42:08 | by: bookofproofs | references: [696]