Let I C R be a proper, prime ideal of R. (a) Let T = RI. Show that T multiplicative set, that is, that 1 ET and if s‚r Є T, then sr ET. We often denote the localization T-¹R by R₁, and call this the "ring R localized at I". (b) Show that m := {iЄI,tЄT} is an ideal of R₁. (c) Show that m is the unique maximal ideal of R. (We say that R, is a local ring in this case, which is just a ring with a unique maximal ideal.)
Let I C R be a proper, prime ideal of R. (a) Let T = RI. Show that T multiplicative set, that is, that 1 ET and if s‚r Є T, then sr ET. We often denote the localization T-¹R by R₁, and call this the "ring R localized at I". (b) Show that m := {iЄI,tЄT} is an ideal of R₁. (c) Show that m is the unique maximal ideal of R. (We say that R, is a local ring in this case, which is just a ring with a unique maximal ideal.)
Elements Of Modern Algebra
8th Edition
ISBN:9781285463230
Author:Gilbert, Linda, Jimmie
Publisher:Gilbert, Linda, Jimmie
Chapter6: More On Rings
Section6.1: Ideals And Quotient Rings
Problem 4E: Exercises
If and are two ideals of the ring , prove that is an ideal of .
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
Transcribed Image Text:Let I C R be a proper, prime ideal of R.
(a) Let T = RI. Show that T multiplicative set, that is, that 1 ET and if
s‚r Є T, then sr ET. We often denote the localization T-¹R by R₁, and call
this the "ring R localized at I".
(b) Show that m := {iЄI,tЄT} is an ideal of R₁.
(c) Show that m is the unique maximal ideal of R. (We say that R, is a local ring
in this case, which is just a ring with a unique maximal ideal.)
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Algebra
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9781285463230
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Gilbert, Linda, Jimmie
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Cengage Learning,