# Product of ideals

From Commalg

## Definition

### Definition with symbols

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

- It is the additive subgroup generated by all elements of the form where
- It is the smallest ideal containing all elements of the form where
- It is the ideal defined as the set of elements of the form:

with

## 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 , denoted , is the subgroup generated by elements of the form where for every - We can also use this to define the notion of power of an ideal. For an ideal , the ideal is simply written times. It is the ideal generated by -fold products of elements from , and need not be the same as the ideal generated by powers of elements from
- In general, the set of products of elements from and is not additively closed. An important exception is the situation where either or is a principal ideal.
- The product of two ideals is contained in their intersection, and contains the square of their intersection. In symbols: