Moist static energy

2013-04-03T01:04:42Z

69.141.157.107: Changed "kinetic Energy" to "Internal Energy" to make clear that the static energy does not include the macroscopic kinetic energy (that's why it's called "static"). Also added "approximately" in conservation statement (neglecting exchange with K.E.).

In [[commutative algebra]], a '''Zariski ring''' is a commutative [[Noetherian ring|Noetherian]] topological ring ''A'' whose topology is defined by an ideal ''m'' contained in the [[Jacobson radical]], the intersection of all maximal ideals. They were introduced by {{harvs|txt|first=Oscar|last=Zariski|year=1946|authorlink=Oscar Zariski}} under the name "semi-local ring" which now means something different, and named "Zariski rings" by {{harvtxt|Samuel|1953}}. Examples of Zariski rings are noetherian local rings and <math>\mathfrak a</math>-adic completions of noetherian rings.

Let ''A'' be a noetherian ring and <math>\widehat{A}</math> its <math>\mathfrak a</math>-adic completion. Then the following are equivalent.
* <math>\widehat{A}</math> is [[faithfully flat module|faithfully flat]] over ''A'' (in general, only flat over it).
* Every maximal ideal is closed for the <math>\mathfrak a</math>-adic topology.
* ''A'' is a Zariski ring.

==References==

* M. Atiyah, I. Macdonald ''Introduction to commutative algebra'' Addison-Wesley Publishing Co., Reading, Mass.-London-Don Mills, Ont. 1969
*{{citation|mr=0054995|last=Samuel|first= Pierre|title=Algèbre locale|series=Mémor. Sci. Math.|volume= 123|publisher= Gauthier-Villars|place= Paris|year= 1953}}
*{{Citation | last1=Zariski | first1=Oscar | author1-link=Oscar Zariski | title=Generalized semi-local rings | id={{MathSciNet | id = 0022835}} | year=1946 | journal=Summa Brasil. Math. | volume=1 | issue=8 | pages=169–195}}
*{{Citation | last1=Zariski | first1=Oscar | author1-link=Oscar Zariski | last2=Samuel | first2=Pierre | author2-link=Pierre Samuel | title=Commutative algebra. Vol. II | publisher=[[Springer-Verlag]] | location=Berlin, New York | isbn=978-0-387-90171-8 | id={{MathSciNet | id = 0389876}} | year=1975}}

[[Category:Commutative algebra]]

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Moist static energy