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Math Help - Checking answer: is 0 a maximal ideal?

  1. #1
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    Checking answer: is 0 a maximal ideal?

    Can 0 be a maximal ideal or prime ideal of R? If this so, what does it say about R in either case ?

    This is my work so far

    Claim: 0 is a prime ideal if R is an integral domain
    Because 0 is a prime if and only if 0|ab implies 0|a or 0|b
    That is : ab = 0 ----> a = 0 or b = 0
    Hence R is an integral domain
    Is that correct?

    And 0 is a maximal ideal. I am not really sure about this. I think the answer is yes if we say R = Q (rational numbers) but I don't know how to explain.

    Thank you very much
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  2. #2
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    Quote Originally Posted by knguyen2005 View Post
    Can 0 be a maximal ideal or prime ideal of R? If this so, what does it say about R in either case ?

    This is my work so far

    Claim: 0 is a prime ideal if R is an integral domain
    Because 0 is a prime if and only if 0|ab implies 0|a or 0|b
    That is : ab = 0 ----> a = 0 or b = 0
    Hence R is an integral domain
    Is that correct?

    And 0 is a maximal ideal. I am not really sure about this. I think the answer is yes if we say R = Q (rational numbers) but I don't know how to explain.

    Thank you very much

    Your claim is correct, and about maximal: {0} is a maximal ideal of R iff R/\{0\} is a field, but...

    Tonio
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  3. #3
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    Quote Originally Posted by tonio View Post
    Your claim is correct, and about maximal: {0} is a maximal ideal of R iff R/\{0\} is a field, but...

    Tonio
    About the maximal ideal,
    So, {0} is a maximal ideal of R iff R/\{0\} is a field. Then we can take R = IR, Q(rational number), C (Complex number). Is that what you are saying.

    Thanks Tonio
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  4. #4
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    Quote Originally Posted by knguyen2005 View Post
    About the maximal ideal,
    So, {0} is a maximal ideal of R iff R/\{0\} is a field. Then we can take R = IR, Q(rational number), C (Complex number). Is that what you are saying.

    Thanks Tonio

    What I was saying is that R/\{0\} \sim R, and thus R is a field.
    It can be of characteristic zero, as the ones you mentioned, of a finite or infinite field of finite positive characteristic.

    Tonio
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