# 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.