Definition 2

Let $A'$ be a ring and $G$ a group operating on $A'$ ($§ 1$, no. 9). Given a prime ideal $p' \subset A'$ the subgroup of elements $\sigma \in G$ such that $\sigma.p' = p'$ is called the decomposition group of $p'$ (with respect to $G$) and is denoted by $G^z(p')$. The ring of elements of $A'$ invariant under $G^z(p')$ is called the decomposition ring of $p'$ (with respect to $G$) and is denoted by $A^z(p')$ (*).

We often write $G^z$ and $A^z$ instead of $G^z(p')$ and $A^z(p')$ respectively, when there is no ambiguity.

For all $\sigma \in G^z(p')$ we also denote by $z \mapsto \sigma.z$ the endomorphism of the ring $A'/p'$ derived from the endomorphism $x \mapsto \sigma.x$ of $A'$ by taking quotients; clearly the group $G^z(p')$ operates in this way on the ring $A'/p'$.