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]

1.Explanation: Characteristic Zero Instead of Characteristic Infinite 

2.Lemma: Any Positive Characteristic Is a Prime Number