p-ring
PRing
2005-05-06 15:07:30
2005-05-06 15:07:30
Definition
strict p-ring
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\begin{defn}
Let $R$ be a commutative ring with identity element equipped with a topology defined by a decreasing sequence:
$$\ldots \subset \mA_3 \subset \mA_2 \subset \mA_1$$
of ideals such that $\mA_n\cdot \mA_m \subset \mA_{n+m}$. We say that $R$ is a $p$-ring if the following conditions are satisfied:
\begin{enumerate}
\item The residue ring $\overline{k}=R/\mA_1$ is a perfect ring of characteristic $p$.
\item The ring $R$ is Hausdorff and complete for its topology.
\end{enumerate}
\end{defn}
\begin{defn}
A $p$-ring $R$ is said to be strict (or a $p$-adic ring) if the topology is defined by the $p$-adic filtration $\mA_n=p^nR$, and $p$ is not a zero-divisor of $R$.
\end{defn}
\begin{exa}
The prototype of strict $p$-ring is the ring of \PMlinkname{$p$-adic integers}{PAdicIntegers} $\Ints_p$ with the usual profinite topology.
\end{exa}
\begin{thebibliography}{9}
\bibitem{serre} J. P. Serre, {\em Local Fields},
Springer-Verlag, New York.
\end{thebibliography}