Product of ideals

Definition with symbols
Suppose $$I,J$$ are ideals in a commutative unital ring $$R$$. Then the product of ideals $$I$$ and $$J$$, denoted $$IJ$$, is defined in the following equivalent ways:


 * It is the additive subgroup generated by all elements of the form $$ab$$ where $$a \in I, b \in J$$
 * It is the smallest ideal containing all elements of the form $$ab$$ where $$a \in I, b \in J$$
 * It is the ideal defined as the set of elements of the form:

$$\sum_{i=1}^n a_ib_i$$

with $$a_i \in I, b_i \in J$$

Facts

 * Product of ideals is commutative and associative. Hence, we can talk of the product of more than two ideals by simply writing them as a string. The product of ideals $$I_1, I_2, \ldots, I_n$$, denoted $$I_1I_2\ldots I_n$$, is the subgroup generated by elements of the form $$a_1a_2\ldots a_n$$ where $$a_j \in I_j$$ for every $$j$$
 * We can also use this to define the notion of power of an ideal. For an ideal $$I$$, the ideal $$I^n$$ is simply $$II \ldots I$$ written $$n$$ times. It is the ideal generated by $$n$$-fold products of elements from $$I$$, and need not be the same as the ideal generated by $$n^{th}$$ powers of elements from $$I$$
 * In general, the set of products of elements from $$I$$ and $$J$$ is not additively closed. An important exception is the situation where either $$I$$ or $$J$$ is a principal ideal.
 * The product of two ideals is contained in their intersection, and contains the square of their intersection. In symbols:

$$(I \cap J)^2 \le IJ \le I \cap J$$