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Math Help - prime ideals and maximals

  1. #1
    Newbie
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    prime ideals and maximals

    Let
    R ={
    a b
    0 a| a, b Z}

    ��
    Then
    R is a ring with unity. Show that

    P
    ={

    0
    b

    0 0
    | b Z}

    is a prime ideal which is not maximal. Carefully justify all your claims.
    [Hint:
    Use the mapping ψ : R Z given by

    ψ{

    a b
    0
    a}

    = a.]
    a b
    0 a are all matrices
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  2. #2
    Senior Member JaneBennet's Avatar
    Joined
    Dec 2007
    Posts
    293
    Quote Originally Posted by Ryan0710 View Post
    Let
    R =\left\{\begin{pmatrix}a & b\\0&a\end{pmatrix}:a,b\in\mathbb{Z}\right\}

    ��
    Then
    R is a ring with unity. Show that

    P =\left\{\begin{pmatrix}0 & b\\0&0\end{pmatrix}:b\in\mathbb{Z}\right\}

    is a prime ideal which is not maximal. Carefully justify all your claims.

    Hint:
    Use the mapping \psi:R\to\mathbb{Z} given by

    \psi\begin{pmatrix}a & b\\0 & a\end{pmatrix}=a
    P is the kernel of the homomorphism \psi; hence the quotient ring R/P is isomorphic to \psi(R)=\mathbb{Z}. Since \mathbb{Z} is an integral domain, so is R/P, i.e. P is a prime ideal.

    Note that Q=\left\{\begin{pmatrix}2a & b\\0&2a\end{pmatrix}:a,b\in\mathbb{Z}\right\} is an ideal of R such that P\subsetneq Q\subsetneq R. Hence P is not maximal in R.

    Thus while every maximal ideal is prime, this example shows that the converse is not true in general. (It’s true in a principal-ideal domain, but R here is not a PID.)
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