7. Let R be an integral domain. a) Show that if R is normal, then so is S-R for any multiplicatively closed set S. b) Show that if the localization Rm at every maximal ideal m c R is normal, then R is normal. c) Show that the localization R, at an element a is isomorphic to the ring R[T]/(aT – 1), and interpret this statement geometrically. Is the assump- tion that R be an integral domain crucial for this statement?

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