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Math Help - Nature of Finitely Generated Ideals

  1. #1
    Super Member Bernhard's Avatar
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    Nature of Finitely Generated Ideals

    I am a math Hobbyist reading Hungerford's book - Abstract Algebra (2nd Edition). In Chapter 6 on Ideals and Quotient Rings he writes (page 138 - see attached copies of pages 137 and 138 for context - whch is Finitely Generated Ideals):

    "Theorem 6.3 : Let R be a commutative ring with identity and c_{1}, c_{2}, ... ... , c_{n}  \in R.

    Then the set I = { r_{1} c_{1} + r_{2} c_{2} + ... ... r_{n} c_{n} | r_{1}, r_{2}, ... ... r_{n},  \in R} is an ideal in R"

    I am trying to get a sense of the elements in the ideal in the Theorem above.

    Are the elements always the sum of n non-zero elements and thus strictly of the form r_{1} c_{1} + r_{2} c_{2} + ... ... r_{n} c_{n} with no terms zero, or does the ideal contain (as I suspect) terms of the form r_{1} c_{1} with all other terms of the sum zero, r_{2} c_{2} + r_{1} c_{1} with all other terms zero, etc as well as the terms containing the sum of n elements.


    Bernhard

    Latex note: in the sum r_{2} c_{2} + r_{1} c_{1} I tried to make the subscripts 2 and 7 but for some reason Latex came up with an error every time I tried subscripts greater than 2!
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  2. #2
    Super Member Bernhard's Avatar
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    Re: Nature of Finitely Generated Ideals

    Forgot attachments - trying to add them.
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    Re: Nature of Finitely Generated Ideals

    suppose a,b are typical elements in I, so that

    a = r_1c_1 + r_2c_2 +\dots + r_nc_n, b = s_1c_1 + s_2c_2 +\dots + s_nc_n.

    then  a-b = (r_1-s_1)c_1 + (r_2-s_2)c_2 +\dots + (r_n-s_n)c_n \in I, so (I,+) is a subgroup of (R,+).

    also, for any d \in R, da = (dr_1)c_1 + (dr_2)c_2 +\dots + (dr_n)c_n \in I, so I is in fact an ideal.

    indeed we can choose r_i = 1, r_j = 0, i \neq j so each c_i \in I.

    on the other hand, any ideal containing \{c_1,c_2,\dots ,c_n\} has to contain every R-linear combination of the c_i,

    by closure of addition, and the multiplicative property of an ideal. so I is the smallest ideal containing the c_i.

    (it should be obvious that 0 terms are allowed. every ideal must contain 0, right?)
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    Super Member Bernhard's Avatar
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    Re: Nature of Finitely Generated Ideals

    Thanks Deveno

    Working by myself I sometimes need the reasurrance of someone confirming these things. Thanks again. Most helpful.
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    Super Member Bernhard's Avatar
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    Re: Nature of Finitely Generated Ideals

    In his example on Page 138 Hungerford writes:

    "In the ring Z[x], the ideal generated by the polynomial x and the constant polynomial 2 consists of all polynomials of the form:

    f(x)x + g(x)2

    It can be shown that this ideal is the ideal I of all polynomials with even constant term."


    But if we take g(x) to be the zero polynomial we are left with polynomials of the form f(x)x (e.g (x + 1)x ) which do not have an even constant term. Am I missing something. {this example started my concern that no terms in the sum could be zero}

    Bernhard
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    Re: Nature of Finitely Generated Ideals

    0 (as a constant) is even.
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