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Math Help - Two Maximal Ideals Proofs

  1. #1
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    Two Maximal Ideals Proofs

    Hello, I am stuck with these questions

    1. Show that \mathbb{Z} \oplus  2\mathbb{Z} is a maximal ideal in \mathbb{Z} \oplus  \mathbb{Z}

    2. Let R be a commutative ring with unity. Prove that every prime ideal of R is also a maximal ideal of R.

    Thanks in advance for your help.
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  2. #2
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    Hi

    1) Assume some ideal J of \mathbb{Z}\oplus\mathbb{Z} strictly contains you ideal, i.e. there is an element in J-\mathbb{Z}\oplus 2\mathbb{Z}. Can you prove that the unity belongs to J .

    2) That's wrong. Take an integral domain which is not a field. Then \{0\} is a prime ideal which is not maximal. A less extreme case: (X) \subset \mathbb{Z}[X] is a prime ideal but not maximal (why?).

    Were there other hypotheses or was it maximal implies prime (which is true)?
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  3. #3
    Super Member Gamma's Avatar
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    Gotta be a PID for 2) to be true.

    But yeah for 1) you could just take that quotient and it is \mathbb{Z}_2 which is a field, so that ideal is maximal.
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