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 fKŒxŁ is a maximal ideal in KŒxŁ if, and only if, f is irreducible. (c) Conclude that Z n D Z =n Z is a ﬁeld if, and only if, ˙ n is prime, and that KŒxŁ=.f / is a ﬁeld if, and only if, f is irreducible. 6.3.8. If J is an ideal of the ring R , show that JŒxŁ is an ideal in RŒxŁ and furthermore RŒxŁ=JŒxŁ Š .R=J/ŒxŁ . Hint: Find a natural homomorphism from RŒxŁ onto .R=J/ŒxŁ with kernel JŒxŁ ....
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 Fall '08
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 Algebra, Commutative ring, nonzero element

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