Power of an ideal

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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.