# What does it mean: the product of two ideals I

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 $a\in I$ and $b\in J$?
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Terry Ray
One would like the product ideal to be:
$IJ=\left\{ij\mid i\in I,j\in J\right\}$
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 $i\prime \prime \in I$, $j\prime \prime \in J$ such that ij+i′j′=i′′j′′ so that its
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Mollie Nash
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 $i\in I$,$j\in J$. (Note that if one of the ideals is principal one can't get such an example.)