Definition 23.3.1. A *divided power ring* is a triple $(A, I, \gamma )$ where $A$ is a ring, $I \subset A$ is an ideal, and $\gamma = (\gamma _ n)_{n \geq 1}$ is a divided power structure on $I$. A *homomorphism of divided power rings* $\varphi : (A, I, \gamma ) \to (B, J, \delta )$ is a ring homomorphism $\varphi : A \to B$ such that $\varphi (I) \subset J$ and such that $\delta _ n(\varphi (x)) = \varphi (\gamma _ n(x))$ for all $x \in I$ and $n \geq 1$.

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