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p-ring (Definition)
Definition 1   Let $ R$ be a commutative ring with identity element equipped with a topology defined by a decreasing sequence:
$\displaystyle \ldots \subset \mathfrak{A}_3 \subset \mathfrak{A}_2 \subset \mathfrak{A}_1$
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. The residue ring $ \overline{k}=R/\mathfrak{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 $ \mathfrak{A}_n=p^nR$, and $ p$ is not a zero-divisor of $ R$.
Example 1   The prototype of strict $ p$-ring is the ring of $ p$-adic integers $ \mathbb{Z}_p$ with the usual profinite topology.

Bibliography

1
J. P. Serre, Local Fields, Springer-Verlag, New York.



"p-ring" is owned by alozano.
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Other names:  $p$-ring, p-adic ring, $p$-adic ring, strict $p$-ring
Also defines:  strict p-ring

Attachments:
Witt vectors (Definition) by alozano
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Cross-references: profinite topology, filtration, strict, complete, Hausdorff, characteristic, perfect ring, ring, residue, ideals, sequence, decreasing, topology, identity element, commutative ring
There is 1 reference to this entry.

This is version 1 of p-ring, born on 2005-05-06.
Object id is 7016, canonical name is PRing.
Accessed 3188 times total.

Classification:
AMS MSC: 13J10 (Commutative rings and algebras :: Topological rings and modules :: Complete rings, completion)
 13K05 (Commutative rings and algebras :: Witt vectors and related rings)
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