(´･ω･`)

Have some random commutative algebra facts


Proposition: A local ring (A,𝔪) has no idempotents ≠ 0, 1.

Proof: Suppose x is an idempotent x ≠ 0, 1. Then x and 1-x are non-units since x(1-x) = x-x² = x-x = 0. Hence they are contained in the maximal Ideal 𝔪 ⊂ A. Thus 1 = x + (1-x) ∈ 𝔪, which is absurd. Hence x can't be idempotent. □