Definition 1.

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


of ideals such that AnAmAn+m. We say that R is a p-ring if the following conditions are satisfied:

  1. 1.

    The residue ring k¯=R/𝔄1 is a perfect ring of characteristicPlanetmathPlanetmath p.

  2. 2.

    The ring R is Hausdorff and completePlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath 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 filtrationPlanetmathPlanetmath An=pnR, 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) p with the usual profinite topology.


Title p-ring
Canonical name Pring
Date of creation 2013-03-22 15:14:28
Last modified on 2013-03-22 15:14:28
Owner alozano (2414)
Last modified by alozano (2414)
Numerical id 4
Author alozano (2414)
Entry type Definition
Classification msc 13J10
Classification msc 13K05
Synonym p-ring
Synonym p-adic ring
Synonym p-adic ring
Synonym strict p-ring
Defines strict p-ring