| Ranked Score: | 750,771,688 | |
| Play Count: | 1,894 | |
| Play Time: | 53h | |
| Max Combo: | 1,725x | |
| Total Hits: | 1,113,010 | |
| Hits x Play: | 588 | |
| Replays Watched: | 1 |
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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. □
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. □
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