Jaffard ring

In mathematics, a Jaffard ring is a type of ring, more general than a Noetherian ring, for which Krull dimension behaves as expected in polynomial extensions. They are named for Paul Jaffard who first studied them in 1960.

Formally, a Jaffard ring is a ring R such that the polynomial ring

$\dim R[T_{1},\ldots ,T_{n}]=n+\dim R,\,$

where "dim" denotes Krull dimension. A Jaffard ring that is also an integral domain is called a Jaffard domain.

The Jaffard property is satisfied by any Noetherian ring R, so examples of non-Jaffardian rings are quite difficult to find. Nonetheless, an example was given in 1953 by Abraham Seidenberg: the subring of

${\overline {\mathbf {Q} }}[[T]]$

consisting of those formal power series whose constant term is rational.

References

• Bouvier, Alain; Kabbaj, Salah (1988). "Examples of Jaffard domains". J. Pure Appl. Algebra. 54 (2–3): 155–165. doi:10.1016/0022-4049(88)90027-8. Zbl 0656.13011.
• Jaffard, Paul (1960). Théorie de la dimension dans les anneaux de polynômes. Mém. Sci. Math. (in French). Vol. 146. Zbl 0096.02502.
• Seidenberg, Abraham (1953). "A note on the dimension theory of rings". Pacific J. Math. 3: 505–512. doi:10.2140/pjm.1953.3.505. ISSN 0030-8730. MR 0054571. Zbl 0052.26902.

External links

• Jaffard ring at PlanetMath.