Unformatted text preview: where the p i are distinct primes. Show that T p a i i ( G ) is the unique p iSylow of G . (b) Show that T p a i i ( G ) is cyclic. (c) Deduce that G is cyclic. (d) Let F be a Fnite Feld. Show that F \ { } is a cyclic group under multiplication. 20 pts. 7. Let R be a commutative ring with 0 n = 1. Let P be a prime ideal of R and S := R \ P . (a) Show that S is a submonoid of ( R, · ). (b) Consider the ring of fractions S − 1 R . Let I := { x s ∈ S − 1 R : x ∈ P, s ∈ S } . Show that I is the unique maximal ideal of S − 1 R . 12 pts. 8. (a) DeFne the notion of projective module. (b) Let d be a divisor of n . Describe a nontrivial Z nmodule structure on Z d . Explain why it is welldeFned. (c) Suppose gcd( d, n/d ) = 1. Show that the Z nmodule Z d is projective. (d) Show that the Z 4module Z 2 is not projective. 20 pts. 1...
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 Spring '08
 comech
 pts, Prime number, Commutative ring

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