Common use of Proposition 1 Clause in Contracts

Proposition 1. 5.8. Let R be a left-artinian ring with ▇▇▇▇▇▇▇▇ radical J(R). Then the natural projection p : R → R/ J(R) induces a surjective map on the set of idempotents. Proof. Let E ∈ R be an idempotent. Then certainly p(E) is an idempotent in R/ J(R). Suppose e ∈ R/ J(R) is an idempotent, i.e. e2 − e ∈ J(R). What we want to find is an element satisfying x2 − x = 0 in R, which is mapped to e. Consider the polynomial F (x) = 3x2 − 2x3. Let e1 := F (e). Then 1 e2 − e1 = (3e2 − 2e3)2 − (3e2 − 2e3) = (4e2 − 4e − 3)(e2 − e)2 ∈ J(R)2, 1 so e2 − e1 ∈ J(R)2. Moreover, e1 = e − (2e − 1)(e2 − e), so e1 ≡ e mod J(R). We define ei := F (e i−1 ). By induction, we have e2 − ei ∈ J(R)2i and e ≡ e i mod J(R). Since R is left-artinian, J(R) is nilpotent, so there exists n ∈ Z≥0 such that e2 − en ∈ J(R)n = 0. Then E = en is the element we were after. Remark 1.5.9. The key to the above proof is that e2 e is nilpotent. Hence we can use the same lifting technique against any nil ideal of R.