Jaffard ring

Let dim denote Krull dimension. A Jaffard ring is a ring A for which dim(A[x])=dim(A)+1 (compare to the bound on the Krull dimension of polynomial rings). Such a ring is said to be Jaffardian.

Since this condition holds for Noetherian ringsMathworldPlanetmath, every Noetherian ring is Jaffardian. Examples of rings that are not Jaffardian are thus relatively difficult to come by, since we are already forced to search exclusively in the realm of non-Noetherian rings. The first example of a non-Jaffardian ring seems to have been found by A. Seidenberg [Seid]: the subring of ¯[[T]] consisting of power series whose constant term is rational.

A Jaffard domain is a Jaffard ring which is also an integral domainMathworldPlanetmath.


  • Seid A. Seidenberg, A note on the dimension theory of rings. Pacific J. of Mathematics, Volume 3 (1953), 505-512.
Title Jaffard ring
Canonical name JaffardRing
Date of creation 2013-03-22 15:22:15
Last modified on 2013-03-22 15:22:15
Owner mathcam (2727)
Last modified by mathcam (2727)
Numerical id 4
Author mathcam (2727)
Entry type Definition
Classification msc 13F05
Defines Jaffard domain
Defines Jaffard