Understanding what $I \cdot J$ means when $I$ and $J$ are ideals.

Question

I have the following definition

For ideals $I, J$ of a ring $R$, their product $I J$ is defined as the ideal of $R$ generated by the elements of the form $x y$ where $x \in I$ and $y \in J$.

Does this mean that $$ IJ = \langle S \rangle $$ where $S = \{xy : x \in I, y \in J \}$?

If this is the case then am struggling with something that according to other questions on this website is obvious. People seem to assert that $IJ \subset I \cap J$ but to me this is not obvious. According to my definitions:

Defn (Generator of ideal) For $A \subseteq R$ a subset, the ideal generated by $A$ is $$ (A)=\left\{\sum_{a \in A} r_a \cdot a: r_a \in R, \text{ only finitely-many non-zero }\right\}. $$

I do not see how the containment is trivially true then. Are my definitions wrong or am I missing something obvious?

Answer 1

You are entirely right about $IJ$. It is equal to $\langle S\rangle$, just as you described.

As for the containment, can you see that every single element of $S$ is contained in $I$? Can you see that every single element of $S$ is contained in $J$? Thus by definition of intersection, $S\subseteq I\cap J$. Can you see why this implies $\langle S\rangle\subseteq I\cap J$?

Answer 2

More formally you have

• $I\cdot J\subset I\cdot A\subset I$, and
• $I\cdot J\subset A\cdot J\subset J$.

Putting the two things together you have $I\cdot J\subset I\cap J$.