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 $a\in I$ and $b\in J$ ?

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Terry Ray

Answered 2021-12-11
Author has **50** answers

One would like the product ideal to be:

$IJ=\{ij\mid i\in I,j\in J\}$

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

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

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$i\in I$ ,$j\in J$ . (Note that if one of the ideals is principal one can't get such an 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

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