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Math Help - Maximal Ideal Question

  1. #1
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    Maximal Ideal Question

    Suppose that R is a commutative ring and |R|=30. If I is an ideal of R and |I|=10, prove that I is a maximal ideal.

    Im pretty sure I understand why its maximal, its because if there was an ideal that properly contained I then the order would be greater than 10 but the order of all proper sub-rings of R would be positive divisors which stop at 10. ok my problem is proving this elegantly on paper... any thoughts. thanks
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  2. #2
    Senior Member Tinyboss's Avatar
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    Well, there's also 15, but you can rule that out, because...?
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  3. #3
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    Quote Originally Posted by fizzle45 View Post
    Suppose that R is a commutative ring and |R|=30. If I is an ideal of R and |I|=10, prove that I is a maximal ideal.

    Im pretty sure I understand why its maximal, its because if there was an ideal that properly contained I then the order would be greater than 10 but the order of all proper sub-rings of R would be positive divisors which stop at 10. ok my problem is proving this elegantly on paper... any thoughts. thanks

    More eleganty than the above? You can remark that R is an (additive) group and an ideal is a subgroup of it so Lagrange's Theorem applies...which ammounts to what you said.

    Tonio
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