What does it mean: the product of two ideals I

Tiffany Russell 2021-12-10 Answered
What does it mean: the product of two ideals I and J is the set of all finite sums of elements of the form ab with aI and bJ?
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Terry Ray
Answered 2021-12-11 Author has 50 answers
One would like the product ideal to be:
IJ={ijiI,jJ}
However, we can easily see that there is a problem. It must be closed under addition, so ij+i′j′ must be in IJ. Can you find iI, jJ such that ij+i′j′=i′′j′′ so that its
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Mollie Nash
Answered 2021-12-12 Author has 33 answers
For example:
I=(2,X) and J=(3,X) in Z[X]. Then IJ=(6,X). Thus XIJ and X can't be written as ij with iI,jJ. (Note that if one of the ideals is principal one can't get such an example.)
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