TITLE: Explaining the product of two ideals
QUESTION [35 upvotes]: My textbook says that 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 $b \in J$. What does this mean exactly? Can you give examples?

REPLY [17 votes]: Since it was also asked for examples, let me mention how to compute the product of two ideals (beyond the already mentioned principal ideals).
If $I$ is generated by elements $\{a_i\}$ and $J$ is generated by elements $\{b_j\}$, then $I \cdot J$ is generated by the elements $\{a_i \cdot b_j\}$. You can verify this either using the element definition of $I \cdot J$, or using the more elegant definition of $I \cdot J$ as the smallest ideal containing all products.
For example, in $\mathbb{Q}[x,y]$, one computes $(x,y) \cdot (x^2,y^2)=(x^3,x y^2,x^2 y,y^3)$. 
In general, one observes that $I \cdot J \subseteq I \cap J$. This is not an equality in general; in the above example the intersection is just $(x,y)$.  However, one has (in the commutative case) $\sqrt{I \cdot J} = \sqrt{I \cap J}$.