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Math Help - principal and primary ideals

  1. #1
    Junior Member
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    principal and primary ideals

    Hello,

    Kindly Can you help me in answer this problem:

    Let Z[X] be the ring of polynomilas of integers

    i) Show that J= 2Z[X]+xZ[X] is not principal

    ii) Show that I=4Z[X]+xZ[X] is a primary ideal of Z[X]
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  2. #2
    Senior Member Tinyboss's Avatar
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    For (i), recognize J as the set of polynomials with even constant term. Any principal generator has to divide everything in J, which in particular contains all the constant polynomials {2, 4, 6, ...}. The only possibilities are 1 and 2. Since J is not the whole ring, 1 is not a generator. And 2 doesn't generate all of J, since it generates polynomials with all even coefficients.

    For (ii), a hint: describe I similarly to how we described J above. If pq is in I, then what do you know about the constant terms of p and q?
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  3. #3
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    Thaaaaaank

    Thaaaaaank you very much for your help I understood well the first question and regarding the second I think that we can describe I as the set of all polynomials in z[x] with multiple of 4 constant term. In other word I is contained in J above. Then if pq is in I, that's means the constant term of pq is multiple of 2, but the constant term is actually the product of constant terms of p and q. So if 2 does not divide the constant term of p, it must devide the constant term of q.

    Thank you again and if there is any mistake or comments, please guide me
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