Let \((R, +,\cdot)\) be a ring free of zero divisors and let \(1\) be its multiplicative and \(0\) its additive identity. Then the minimal^{1} 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 \[\operatorname{char}( R ):=n.\]
Explanations: 1
Definitions: 1
Explanations: 2
Lemmas: 3
Proofs: 4
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. ↩