Idempotents in ring extensions

The aim of the paper is to study idempotents of ring extensions R⊆SR⊆S where S   stands for one of the rings R[x1,x2,…,xn]R[x1,x2,…,xn], R[x1±1,x2±1,…,xn±1], R〚x1,x2,…,xn〛R〚x1,x2,…,xn〛. We give criteria for an idempotent of S to be conjugate to an idempotent of R. Using our criteria we show, in particular, that idempotents of the power series ring are conjugate to idempotents of the base ring and we apply this to give a new proof of the result of P.M. Cohn (2003) [4, Theorem 7] that the ring of power series over a projective-free ring is also projective-free. We also get a short proof of the more general fact that if the quotient ring R/JR/J of a ring R by its Jacobson radical J is projective-free then so is the ring R.