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Concept# Divided power structure

Summary

In mathematics, specifically commutative algebra, a divided power structure is a way of making expressions of the form x^n / n! meaningful even when it is not possible to actually divide by n!.
Definition
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 \gamma_n : I \to A for n = 0, 1, 2, ... such that:
#\gamma_0(x) = 1 and \gamma_1(x) = x for x \in I, while \gamma_n(x) \in I for n > 0.
#\gamma_n(x + y) = \sum_{i=0}^n \gamma_{n-i}(x) \gamma_i(y) for x, y \in I.
#\gamma_n(\lambda x) = \lambda^n \gamma_n(x) for \lambda \in A, x \in I.
#\gamma_m(x) \gamma_n(x) = ((m, n)) \gamma_{m+n}(x) for x \in I, where ((m, n)) = \frac{(m+n)!}{m! n!} is an integer.
#\gamma_n(\gamma_m(x)) = C_{n, m

Official source

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