# p-ring

###### Definition 1.

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

 $\ldots\subset\mathfrak{A}_{3}\subset\mathfrak{A}_{2}\subset\mathfrak{A}_{1}$

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

1. 1.

The residue ring $\overline{k}=R/\mathfrak{A}_{1}$$p$.

2. 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 $\mathfrak{A}_{n}=p^{n}R$, 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.

## References

• 1 J. P. Serre, , Springer-Verlag, New York.
Title p-ring Pring 2013-03-22 15:14:28 2013-03-22 15:14:28 alozano (2414) alozano (2414) 4 alozano (2414) Definition msc 13J10 msc 13K05 $p$-ring p-adic ring $p$-adic ring strict $p$-ring strict p-ring