Let $ (A, \mathfrak{m}) $ be a local ring and let $ B $ be a finite $A$-algebra. There is a claim in Milne's Étale Cohomology book that if $ B $ is not local, then there is $ b \in B $ such that $ \overline{b} $ is a non-trivial idempotent in $ B/\mathfrak{m} B $.
I've tried to work with the definition of local rings that whenever $ a + b = 1 $ then either $ a $ or $ b $ is invertible but I have not succeeded in proving this claim. How do I prove this?