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Show this need NOT be an ideal

  1. Jul 14, 2008 #1
    Let I,J be ideals of a ring R. Show that the set of products of elements of I,J need not be an ideal (by counterexample - I have been trying to use a polynomial ring).
  2. jcsd
  3. Jul 14, 2008 #2

    matt grime

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    And how has it gone? What examples have you tried, and where has it gone wrong?
  4. Jul 17, 2008 #3
    I think I may have solved it now, and would appreciate confirmation or corrections:

    Let R[x,y] be the ring of polynomials with real coefficients. Let I be the ideal containing all elements of C with zero constant term. Let J be the ideal containing all elements of C with zero constant term and zero coefficients of x, y and xy.

    Then I has multiples of x, y, xy, x^2, y^2, etc.
    And J has multiples of x^2, y^2, x*y^2, y*x^2, etc.

    Now IJ contains x*x^2 = x^3 and y*y^2 = y^3.

    But x^3 + y^3 is factorised uniquely (since to irreducible factors) as (x + y)(x^2 - xy + y^2). Neither of these polynomials is in J, and therefore the sum is not an element of IJ. So IJ is not closed under addition.
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