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Math Help - prime ideals in a product of rings

  1. #1
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    prime ideals in a product of rings

    Let B=A_{1}\times\cdots\times A_{n} be a product of rings. Any prime ideal of B is of the form A_{1}\times\cdots\times I_{i}\times\cdots \times A_{n}, where i\in\{1,\cdots,n\} and I_{i} is a prime ideal of A_{i}.

    Thanks in advance.
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  2. #2
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    Quote Originally Posted by Biscaim View Post
    Let B=A_{1}\times\cdots\times A_{n} be a product of rings. Any prime ideal of B is of the form A_{1}\times\cdots\times I_{i}\times\cdots \times A_{n}, where i\in\{1,\cdots,n\} and I_{i} is a prime ideal of A_{i}.

    Thanks in advance.
    you should always mention what kind of rings do you have? for example, are they commutative? are they unitary? since you didn't mention that, i'll assume that your rings are commutative.

    we first need an important fact:

    Fact: every ideal of B is in the form I=I_1 \times \cdots I_n, where each I_j is an ideal of A_j.

    Proof: it's clear that I is an ideal of B. conversely, suppose I is any ideal of B. for any 1 \leq j \leq n consider the map I \overset{\iota} \longrightarrow B \overset{\pi_j} \longrightarrow A_j, where \iota and \pi_j are the inclusion and the projection maps

    respectively. let I_j=\pi_j \iota(I). see that I_j is an ideal of A_j and I=I_1 \times \cdots \times I_n. Q.E.D.

    now let I=I_1 \times \cdots \times I_n be any ideal of B. we have \frac{B}{I} \cong \frac{A_1}{I_1} \times \cdots \times \frac{A_n}{I_n}. we know that I is prime iff \frac{B}{I} is a domain. now suppose I_i \neq A_i, \ I_j \neq A_j, for some i \neq j. choose a_i \in A-I_i and

    a_j \in A_j - I_j. let x=(0, \cdots, 0, a_i + I_i, 0 , \cdots , 0) and y=(0, \cdots, 0, a_j + I_j, 0 , \cdots , 0). then xy=0 \in \frac{A_1}{I_1} \times \cdots \times \frac{A_n}{I_n} and x \neq 0, y \neq 0. so in this case \frac{B}{I} is not a domain. thus in order for \frac{B}{I}

    to be a domain, we must have I_j=A_j for all but one j, which we'll call it i. then I=A_1 \times \cdots \times I_i \times \cdots \times A_n and \frac{B}{I} \cong \frac{A_i}{I_i}. clearly I is a prime ideal of B iff I_i is a prime ideal of A_i. \ \Box
    Last edited by NonCommAlg; June 28th 2009 at 11:10 AM.
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