Definition 23.2.1. Let $A$ be a ring. Let $I$ be an ideal of $A$. A collection of maps $\gamma _ n : I \to I$, $n > 0$ is called a *divided power structure* on $I$ if for all $n \geq 0$, $m > 0$, $x, y \in I$, and $a \in A$ we have

$\gamma _1(x) = x$, we also set $\gamma _0(x) = 1$,

$\gamma _ n(x)\gamma _ m(x) = \frac{(n + m)!}{n! m!} \gamma _{n + m}(x)$,

$\gamma _ n(ax) = a^ n \gamma _ n(x)$,

$\gamma _ n(x + y) = \sum _{i = 0, \ldots , n} \gamma _ i(x)\gamma _{n - i}(y)$,

$\gamma _ n(\gamma _ m(x)) = \frac{(nm)!}{n! (m!)^ n} \gamma _{nm}(x)$.

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