Power of an ideal
From Commalg
Definition
Let be a commutative unital ring and
be an ideal in
. The
power of
, denoted
, is defined in the following equivalent ways:
- It is the ideal generated by
-fold products of elements from
- It is the product of the ideal
with itself,
times.
In symbols, it is the additive subgroup generated by elements of the form where
.
The second power of an ideal is termed its square, and the third power is termed its cube.
Facts
- For a principal ideal, the
power is the same as the ideal generated by the
power of its generator. However, in general, it may not be true that the
powers of elements of an ideal generate the
power of the ideal.