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Unformatted text preview: 2.7.13 . 6.3.5. Prove Proposition 6.3.10 (the diamond isomorphism theorem for rings) following the pattern of the proof of Proposition 2.7.18 (the diamond isomorphism theorem for groups). 6.3.6. Prove that an ideal M in R is maximal if, and only if, R=M is simple. 6.3.7. (a) Show that n Z is maximal ideal in Z if, and only if, n is a prime. (b) Show that .f / D fKx is a maximal ideal in Kx if, and only if, f is irreducible. (c) Conclude that Z n D Z =n Z is a eld if, and only if, n is prime, and that Kx=.f / is a eld if, and only if, f is irreducible. 6.3.8. If J is an ideal of the ring R , show that Jx is an ideal in Rx and furthermore Rx=Jx .R=J/x . Hint: Find a natural homomorphism from Rx onto .R=J/x with kernel Jx ....
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 Fall '08
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 Algebra

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