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.