Buffon's noodle

2014-02-03T18:11:38Z

128.196.226.219: "number of crossing" to "number of crossings"

In [[mathematics]], a '''Jaffard ring''' is a type of [[ring (mathematics)|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

:<math>\dim R[T_1,\ldots,T_n] = n + \dim R, \,</math>

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

:<math>\overline{\mathbf{Q}} [[T]]</math>

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

==References==
{{reflist}}
* {{cite journal | zbl=0656.13011 | last1=Bouvier | first1=Alain | last2=Kabbaj | first2=Salah | title=Examples of Jaffard domains | journal=J. Pure Appl. Algebra | volume=54 | number=2-3 | pages=155–165 | year=1988 }}
* {{cite book | zbl=0096.02502 | last=Jaffard | first=Paul | title=Théorie de la dimension dans les anneaux de polynômes | language=French | series=Mém. Sci. Math. | volume=146 | year=1960 }}
* {{cite journal | last = Seidenberg | first = Abraham | authorlink = Abraham Seidenberg | title = A note on the dimension theory of rings | journal = Pacific J. Math. | volume = 3 | year = 1953 | pages = 505–512 | issn = 0030-8730 | mr=0054571 | zbl=0052.26902 }}

==External links==

* {{PlanetMath|urlname=JaffardRing|title=Jaffard ring}}

{{Abstract-algebra-stub}}

[[Category:Ring theory]]

formulasearchengine - User contributions [en]

Buffon's noodle