What does it mean: the product of two ideals I

Question

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

?

Answer 1

One would like the product ideal to be:

I J = {i j ∣ 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'' \in I

,

j'' \in J

such that ij+i′j′=i′′j′′ so that its

Answer 2

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.)

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