Unformatted text preview: Recall that we deﬁned localization in general for any multiplicative system S of a commutative ring R , maybe containing zerodivisors. 1 . Let R be a commutative ring and let S be a multiplicative system in R . Let RS1 be the localicalization of R at S . Let I be an ideal of R . (a). Show that IS1 = ± r s  r ∈ I,s ∈ S ² is an ideal of RS1 . Show also that S = { s + I  s ∈ S } is a multiplicative system in the factor ring R/I . Now show that RS1 /IS1 ∼ = ( R/I )( S )1 . (b). Let P be a prime ideal and recall that the localization of R at P is R P = RS1 where S = { x ∈ R  x 6∈ P } . Show that RS1 /PS1 is isomorphic to the ﬁeld of fractions of the domain R/P . 1...
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 Fall '08
 Bunch,J
 Math, Ring, Prime number, Integral domain, Ring theory, R. Let RS

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