Lemma 2

Let $A$ be a commutative algebra over $K$ and $e$ an idempotent of $A$; then we have $de = 0$ in $\Omega_K(A)$.

From the relation $e = e^2$ we deduce $de = 2e \cdot de$; on multiplying by $e$ we obtain $e \cdot de = 2e \cdot de$, hence $e \cdot de = 0$, and so finally, $de = 2e \cdot de = 0$.

C) We first prove two lemmas: