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 C R is normal, 7. 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?
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 C R is normal, 7. 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?
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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part b andc solution
![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?](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F76fbdb8b-7bfc-48c4-99e5-e81330a4cccf%2F1dbda77a-23e5-4b92-a479-1555907dde98%2F9m21xgg_processed.png&w=3840&q=75)
Transcribed Image Text: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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