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

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

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

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

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

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

In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).