Understanding the ideal $IJ$ in $R$

This question already has an answer here:

  • Defining the Product of Ideals

    2 answers

Solutions Collecting From Web of "Understanding the ideal $IJ$ in $R$"

Consider the four-variable polynomial ring $\Bbb C[a,b,c,d]$ and ideals $I=(a,b)$, $J=(c,d)$.

Then $IJ$ contains the elements $ac$ and $bd$ hence contains $ac+bd$, but the set

$$\{xy:x\in I,y\in J\}$$

does not contain $ac+bd$, even though it contains $ac$ and $bd$. So no, this set is not closed under addition, and it is not $IJ$. The problem is that summing products of elements in $I$ and $J$ does not generally yield something that can be simplified to a single such product.

By the way, try proving $\{xy:x\in I,y\in J\}$ doesn’t contain $ac+bd$ as an exercise.