p-ring

Definition 1.

Let $R$ be a commutative ring with identity element equipped with a topology defined by a decreasing sequence:

$$\mathrm{\dots }\subset {𝔄}_3\subset {𝔄}_2\subset {𝔄}_1$$

of ideals such that ${\mathrm{A}}_n\mathrm{\cdot }{\mathrm{A}}_m\mathrm{\subset }{\mathrm{A}}_{n\mathrm{+}m}$. We say that $R$ is a $p$-ring if the following conditions are satisfied:

1. The residue ring $\overline{k}=R/{𝔄}_1$ is a perfect ring of characteristic $p$.

2. The ring $R$ is Hausdorff and complete for its topology.

Definition 2.

A $p$-ring $R$ is said to be strict (or a $p$-adic ring) if the topology is defined by the $p$-adic filtration ${\mathrm{A}}_n\mathrm{=}{p}^n\mathit{}R$, and $p$ is not a zero-divisor of $R$.

Example 1.

The prototype of strict $p$-ring is the ring of $p$-adic integers (http://planetmath.org/PAdicIntegers) ${\mathbb{Z}}_p$ with the usual profinite topology.