Characteristic of a ring

The characteristic is the additive order of 1 when that order is finite, and 0 when 1 has infinite order. So \(\mathbb{Z}_n\) has characteristic \(n\), while \(\mathbb{Z}\), \(\mathbb{Q}\), \(\mathbb{R}\) and \(\mathbb{Z}[x]\) have characteristic 0. In a product \(R\times S\) the characteristic is the least common multiple of those of \(R\) and \(S\), not their product.

If the characteristic is \(n>0\), then \(n\cdot a=(n\cdot1)a=0\) for all \(a\in R\): the number \(n\) annihilates every element, so the additive order of any \(a\) divides \(n\). Thus the characteristic is read off from 1 but governs the whole ring, and in characteristic \(n\) the elements \(k\cdot1\) depend only on \(k\) modulo \(n\).

If a ring without zero divisors had composite characteristic \(n=ab\) with \(1<a,b<n\), then \((a\cdot1)(b\cdot1)=0\) with both factors nonzero, which is impossible. Hence the characteristic of an integral domain is 0 or a prime \(p\); in the latter case the domain contains a copy of \(\mathbb{Z}_p\). Finite fields always have prime characteristic.

A frequent mistake is to confuse the characteristic with the number of elements: the field \(\mathbb{F}_4\) has 4 elements and characteristic 2. This notion will be key for classifying fields: the characteristic is 0 (and they contain \(\mathbb{Q}\)) or a prime \(p\) (and they contain \(\mathbb{Z}_p\)).