Definition:Product of Ideals of Ring

Definition
Let $A$ be a commutative ring.

Let $I,J$ be ideals of $A$.

The product of $I$ and $J$ as ideals is the set of all finite sums:


 * $IJ = \{a_1b_1 + \cdots + a_rb_r : a_i \in I,\ b_i \in J, r \in \N \}$

Note
The product of $I$ and $J$ as ideals is different from their product as subsets.

In fact the former is generated by the latter.