Ring theory MOC

Characteristic

The characteristic char(𝑅) of a rng 𝑅 is the smallest positive integer 𝑛 ∈ℕ such that the sum of 𝑛 copies of any 𝑎 ∈𝑅 is 0, i.e. 𝑛𝑎 =0. #m/def/ring If no such 𝑛 exists then char(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 ∈ℕ such that ∑𝑛1 =0 and thus char(𝑅) =0. Now suppose that 1 has additive order 𝑛, i.e. 𝑛 is the smallest positive integer such that ∑𝑛1 =0 and thus char(𝑅) ≥𝑛. Now for any 𝑥 ∈𝑅

𝑛∑𝑥=𝑛∑1𝑥=(𝑛∑𝑥)1=0⋅1=0

hence char(𝑅) =𝑛.

Properties

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

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