Product of Ideals

Question

Can I define the product of ideals $I, J$ as $$IJ = \left\{\sum_{j}^n \sum_{i}^n a_i b_j| a_i \in I, b_j \in J, i,j \in \mathbb{N}\right\}.$$ Are there some books that define the product of ideals like this? My professor asked me the explanation of product of ideal $$AB=\left\{\sum_{i}^n a_i b_i| a_i \in A, b_i \in B, n \in \mathbb{N}\right\}.$$ And she asked me if this definition means that the number of elements in $A$ are equal to $B$. So, what if the number of elements is different? Could someone explain this? Thanks.

Answer 1

A general element in the product of two ideals $\,I\,,\,J\,$ is a finite sum of elements of the form $\,xy\;,\;\;x\in I\;,\;y\in J\;$...which is exactly what you wrote in your second definition (understanding $\,n\in\Bbb N\,$ is variable, not fixed.

Elements of the form $\,x\;,y\;,\;x+y\;$ etc. , do not usually belong to the product. One way of getting convinced of this is to realize that $\,IJ\subset I\cap J\;$ , so if we had, for example, for$\;x\in I\;;\;\; x\in IJ\,$ , then

$$x\in IJ\subset I\cap J\le J$$

which of course doesn't generally holds.

So yes: the finite sums of products of elements, one in the first ideal and the other in the second one, means that the number of elements taking part in each such a product in each particular element of $\,IJ\,$ is the same.