7. Let R be an integral domain. a) Show that if R is normal, then so is S-¹R for any multiplicatively closed set S. b) Show that if the localization Rm at every maximal ideal m CR 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?

7. Let R be an integral domain. a) Show that if R is normal, then so is S-¹R for any multiplicatively closed set S. b) Show that if the localization Rm at every maximal ideal m CR 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?

Elements Of Modern Algebra

8th Edition

ISBN:9781285463230

Author:Gilbert, Linda, Jimmie

Publisher:Gilbert, Linda, Jimmie

Chapter5: Rings, Integral Domains, And Fields

Section5.4: Ordered Integral Domains

Problem 6E: 6. Prove that if is any element of an ordered integral domain then there exists an element such...

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