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: