Power of an ideal

Definition
Let $$R$$ be a commutative unital ring and $$I$$ be an ideal in $$R$$. The $$n^{th}$$ power of $$I$$, denoted $$I^n$$, is defined in the following equivalent ways:


 * It is the ideal generated by $$n$$-fold products of elements from $$I$$
 * It is the product of the ideal $$I$$ with itself, $$n$$ times.

In symbols, it is the additive subgroup generated by elements of the form $$a_1a_2 \ldots a_n$$ where $$a_i \in I$$.

The second power of an ideal is termed its square, and the third power is termed its cube.

Facts

 * For a principal ideal, the $$n^{th}$$ power is the same as the ideal generated by the $$n^{th}$$ power of its generator. However, in general, it may not be true that the $$n^{th}$$ powers of elements of an ideal generate the $$n^{th}$$ power of the ideal.