Let R be a commutative domain, and let I be a prime ideal ofR.
(i) Show that S defined as R \ I (the complement of I in R) ismultiplicatively closed.
(ii) By (i), we can construct the ring R1 =S-1R, as in the course. Let D = R / I. Show that
the ideal of R1 generated by I, that is,IR1, is maximal, and R1 / I1R isisomorphic to the
field of fractions of D. (Hint: use the fact that everything inS-1R can be written in the
form s-1r, where s ∈ S and r ∈ R. The first step isto show that IR1 ∩ R = I).
