Product of ideals

Definition

Definition with symbols

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

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

$\sum_{i = 1}^{n} a_{i} b_{i}$

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 $I_{1}, I_{2}, \dots, I_{n}$ , denoted $I_{1} I_{2} \dots I_{n}$ , is the subgroup generated by elements of the form $a_{1} a_{2} \dots a_{n}$ 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 $I I \dots I$ 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 $n^{t h}$ 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:

$(I \cap J)^{2} \leq I J \leq I \cap J$