“Zariski ring”的意思、由来-开放百科全书

词条

Zariski ring

释义

References

In commutative algebra, a Zariski ring is a commutative Noetherian topological ring A whose topology is defined by an ideal 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 {{harvs|txt|last=Samuel|first= Pierre|author-link=Pierre Samuel|year=1953}}. Examples of Zariski rings are noetherian local rings with the topology induced by the maximal ideal, and -adic completions of Noetherian rings.

Let A be a Noetherian topological ring with the topology defined by an ideal . Then the following are equivalent.

A is a Zariski ring.
The completion is faithfully flat over A (in general, it is only flat over A).
Every maximal ideal is closed.

References

{{Citation | last1=Atiyah | first1=Michael F. | author1-link=Michael Atiyah | last2=Macdonald | first2=Ian G.|author2-link=Ian G. Macdonald | title=Introduction to commutative algebra | publisher=Addison-Wesley Publishing Co., Reading, Mass.-London-Don Mills, Ont. | mr=0242802 | year=1969}}
{{citation|mr=0054995|last=Samuel|first= Pierre|author-link=Pierre Samuel|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 |mr=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 |mr=0389876 | year=1975}}

1 : Commutative algebra

随便看

开放百科全书收录14589846条英语、德语、日语等多语种百科知识，基本涵盖了大多数领域的百科知识，是一部内容自由、开放的电子版国际百科全书。