Question

10 Let R be a commutative domain, and let I be a prime ideal of R. (i) Show that S defined as R I (the complement of I in R) is multiplicatively closed. (ii) By (i), we can construct the ring Ri = S-1R, as in the course. Let D = R/I. Show that the ideal of R1 generated by 1, that is, I R1, is maximal, and RI/I R is isomorphic to the field of fractions of D. (Hint: use the fact that everything in S-'R can be written in the form s-?r, where s ES and r e R. The first step is to show that IRR = 1).

Answer 1

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Lab XEIR NR = Sxe IR, IVER X EIR, . 52 cutele rei XER = 's is 'R as ER a) IR, OREI A as IR, ARE I casian Maya Ø : - Ri [ IRI x+I -> xt IR, Y is (1-1) if y Cact I) = 4(x2+2) = 1 +28 = 3.42R, as a plq EIR, & X, Xq ER

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