Since this condition holds for Noetherian rings, 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 consisting of power series whose constant term is rational.
A Jaffard domain is a Jaffard ring which is also an integral domain.
- Seid A. Seidenberg, A note on the dimension theory of rings. Pacific J. of Mathematics, Volume 3 (1953), 505-512.
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