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Homework Help: Product Ideal

  1. Oct 27, 2012 #1
    1. The problem statement, all variables and given/known data
    Show that if I and J are ideals of the (commutative) ring; R then
    S={xy|x in I y in J} is not necessarily an ideal but the set of finite sums
    IJ={Ʃ(xvyv)|xvin I yvin J}
    is (and called the product ideal).

    2. Relevant equations
    An ideal satisfies the properties
    For all x, x' in I, all r in R
    (i) x+x' in I
    (ii) rx in I

    3. The attempt at a solution
    This seems wrong to me since if I is generated by i and J is generated by j
    xy+x'y'= ai*bj+a'i*b'j=(ab+a'b')(ij)=(ab+a'b')i*j which is in S since (ab+a'b')i is in I and j is in J...
    and obviously by the same rule;
    r(xy) is in S.
    Therefore, it seems to me that S and IJ are equivalent sets (since the elements of S could be split up into a sum of product elements) and I don't see how S could _not_ be an ideal.

    I haven't been able to find much information about product ideals; but this is a problem in my textbook - Algebra by Michael Artin (second edition) so I'm disinclined to think the lecturer phrased the question wrong...
    Please help?
  2. jcsd
  3. Oct 28, 2012 #2

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