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Math Help - Prime ideal and maximal

  1. #1
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    Prime ideal and maximal

    If R is a finite commutative ring with unity, prove that every prime ideal of R is a maximal.

    Proof. Suppose that <p> is a prime ideal of R, then p is a prime. Let p | ab, we have p|a or p|b. Let <p> \subset J \subset R, I wish to show that J = R.

    What can I use here?
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  2. #2
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    Quote Originally Posted by tttcomrader View Post
    If R is a finite commutative ring with unity, prove that every prime ideal of R is a maximal.

    Proof. Suppose that <p> is a prime ideal of R, then p is a prime. Let p | ab, we have p|a or p|b. Let <p> \subset J \subset R, I wish to show that J = R.

    What can I use here?
    Let \mathfrak{n} be an ideal. If \mathfrak{n} is a prime ideal then R/\mathfrak{n} is an integral domain. But it is a finite integral domain. Thus, R/\mathfrak{n} is a field, thus, \mathfrak{n} must be a maximal ideal. Q.E.D.
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