Integral domain

The characteristic of an integral domain is 0 or prime

Let 𝐷 be an integral domain. Then its characteristic char(𝐷) is either 0 or prime. ring

Proof

Suppose that 1 has order 𝑛 =𝑠𝑡 where 1 ≤𝑠,𝑡 ≤𝑛. Then

0=𝑛∑1=(𝑠∑1)(𝑡∑1)

so either 𝑠 =𝑛 or 𝑡 =𝑛. Thus 𝑛 is prime.

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