Let P be a prime ideal in a ring R (commutative with identity) and I be the ideal generated by the idempotents of R, contained in P. Show that R/I contain no idempotents other than 0 and 1.
An approach suggested is to show that for $x$ being one said idempotent in R/I, Then $x(x-1)$ belongs to I.If we then take X ∈ P , then one way suggested in the hint is to show that ∃ y ∈ I and y^2=y such that $yx(x-1)$=$x(x-1)$. I am unable to find this and then show (1-y)x is idempotent which implies x is idempotent