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200a-f11-hw7 - Recall that we defined localization in...

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Math 200a Fall 2011 Homework 7 Due Tuesday 11/22/2011 by 5pm in homework box on 6th floor of AP&M Upcoming schedule: Note that this homework is not due on Friday 11/18, but rather the following Tuesday. This is because we will have no HW due at the end of Thanksgiving week, and so you get a bit of extra time for this one. Note that class is cancelled on Wednesday, 11/23. There will be one final homework after this one, which will be due the last Friday of class, December 2. Reading assignment: Chapter 8, and sections 9.1 and 9.2. Exercises to be handed in: (all exercise numbers refer to Dummit and Foote, 3rd edition.) Section 7.5: 2 Section 7.6: 5 Section 8.1: 8(a) , (do for D = - 2 , - 3 only), 10 Section 8.2: 4, 6, 8 Section 9.2: 1, 2, 7 Exercise not from the text: (to be handed in):
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Unformatted text preview: Recall that we defined localization in general for any multiplicative system S of a com-mutative ring R , maybe containing zerodivisors. 1 . Let R be a commutative ring and let S be a multiplicative system in R . Let RS-1 be the localicalization of R at S . Let I be an ideal of R . (a). Show that IS-1 = ± r s | r ∈ I,s ∈ S ² is an ideal of RS-1 . Show also that S = { s + I | s ∈ S } is a multiplicative system in the factor ring R/I . Now show that RS-1 /IS-1 ∼ = ( R/I )( S )-1 . (b). Let P be a prime ideal and recall that the localization of R at P is R P = RS-1 where S = { x ∈ R | x 6∈ P } . Show that RS-1 /PS-1 is isomorphic to the field of fractions of the domain R/P . 1...
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