Characteristic

The characteristic $c h a r (𝑅)$ of a rng $𝑅$ is the smallest positive integer $𝑛 \in ℕ$ such that the sum of $𝑛$ copies of any $𝑎 \in 𝑅$ is $0$ , i.e. $𝑛 𝑎 = 0$ . #m/def/ring If no such $𝑛$ exists then $c h a r (𝑅) = 0$ . For a ring with unity, the characteristic is the additive group order of unity (or zero if the order is infinite).

Proof

If $1$ has infinite additive order, then there is no $𝑛 \in ℕ$ such that $\sum 𝑛 1 = 0$ and thus $c h a r (𝑅) = 0$ . Now suppose that $1$ has additive order $𝑛$ , i.e. $𝑛$ is the smallest positive integer such that $\sum 𝑛 1 = 0$ and thus $c h a r (𝑅) \geq 𝑛$ . Now for any $𝑥 \in 𝑅$

𝑛 \sum 𝑥 = 𝑛 \sum 1 𝑥 = (𝑛 \sum 𝑥) 1 = 0 \cdot 1 = 0

hence $c h a r (𝑅) = 𝑛$ .

Properties

The characteristic of an integral domain is 0 or prime
^P1 (this gives a nice alternative definition of characteristic for a ring)

#state/tidy | #lang/en | #SemBr