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Math Help - Prove that A/I has the same characteristic as A.

  1. #1
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    Prove that A/I has the same characteristic as A.

    (a) Suppose that A is an integral domain of positive characteristic. Suppose that I \subsetneq A is an ideal. Prove that A/I has the same characteristic as A.

    (b) Give an example which demonstrates that A/I need not be an integral domain.
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  2. #2
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    Quote Originally Posted by JJMC89 View Post
    (a) Suppose that A is an integral domain of positive characteristic. Suppose that I \subsetneq A is an ideal. Prove that A/I has the same characteristic as A.


    Hints - Try to show the following:

    1) The characteristic of any integral domain is either zero or a prime

    2) If for an abelian group G we define exponent(G):=Exp(G):=\max\{n\in\mathbb{N} | n=ord(x)\mbox{ for some }x\in G\} , then

    the exponent of any subgroup of G is a divisor of the group's exponent or zero if the

    latter is zero.



    (b) Give an example which demonstrates that A/I need not be an integral domain.

    Take A:=\mathbb{F}_p[x]\,,\,\,I=<x^2> , with \mathbb{F}_p = the prime field

    of characteristic p.

    Tonio
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  3. #3
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    1) I did this as a previous proof.

    2) What does the exponent have to do with proving that the quotient ring has the same characteristic as the integral domain?
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  4. #4
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    Quote Originally Posted by JJMC89 View Post
    1) I did this as a previous proof.

    2) What does the exponent have to do with proving that the quotient ring has the same characteristic as the integral domain?


    An ideal is also an additive subgroup of the abelian additive group of the ring, and then

    in the quotient group A/I ...

    Tonio
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  5. #5
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    So you're saying that the exponent of the Abelian additive subgroup of the ring and the characteristic of the ring are the same and that the exponent of the quotient group is the same as the characteristic of the quotient ring?
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  6. #6
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    Quote Originally Posted by JJMC89 View Post
    So you're saying that the exponent of the Abelian additive subgroup of the ring and the characteristic of the ring are the same and that the exponent of the quotient group is the same as the characteristic of the quotient ring?


    I'm saying the former (it's the definition of ring characteristic!), and the latter is what you have to prove...

    Suppose p>0 is the characteristic (exponent) of R , and take 1+I\in R/I .

    Clearly p(1+I)=p\cdot 1+I=0\mbox{ in } R/I\Longrightarrow the exponent of the

    quotient ring has to be a divisor of p, but since this is a prime...

    Tonio
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  7. #7
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    Quote Originally Posted by tonio View Post
    it's the definition of ring characteristic!
    The definition I was given for ring characteristic is different, but now I see that they are the same.



    I've got it now. Thanks.
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