Divided power structure

In mathematics, specifically commutative algebra, a divided power structure is a way of making expressions of the form ${\displaystyle x^{n}/n!}$ meaningful even when it is not possible to actually divide by ${\displaystyle n!}$.

Let A be a commutative ring with an ideal I. A divided power structure (or PD-structure, after the French puissances divisées) on I is a collection of maps ${\displaystyle \gamma _{n}:I\to A}$ for n=0, 1, 2, ... such that:

For convenience of notation, ${\displaystyle \gamma _{n}(x)}$ is often written as ${\displaystyle x^{[n]}}$ when it is clear what divided power structure is meant.

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