In algebra, an '''analytically normal ring''' is a local ring whose completion is a normal ring, in other words a domain that is integrally closed in its quotient field.
{{harvtxt|Zariski|1950}} proved that if a local ring of an algebraic variety is normal, then it is analytically normal, which is in some sense a variation of Zariski's main theorem. {{harvs|txt|last=Nagata|year1=1958|year2=1962|loc2=Appendix A1, example 7}} gave an example of a normal Noetherian local ring that is analytically reducible and therefore not analytically normal.
== References ==
*{{citation|mr=0097395|last=Nagata|first= Masayoshi|title=An example of a normal local ring which is analytically reducible|journal=Mem. Coll. Sci. Univ. Kyoto. Ser. A Math.|volume= 31|year= 1958|pages= 83–85|url= http://projecteuclid.org/euclid.kjm/1250776950}} *{{citation|authorlink=Masayoshi Nagata|last=Nagata|first= Masayoshi|title=Local rings|series= Interscience Tracts in Pure and Applied Mathematics|volume= 13|publisher= Interscience Publishers|place=New York-London |year=1962|isbn= ((978-0470628652))<!-- isbn ok, goes to later reprint of same edition by same publisher -->}} *{{citation| mr=0024158 |last=Zariski|first= Oscar|authorlink=Oscar Zariski|title=Analytical irreducibility of normal varieties|journal=Annals of Mathematics | series = Second Series |volume=49|year=1948|issue=2|pages= 352–361|doi=10.2307/1969284|jstor=1969284}} *{{citation|mr=0045413|last=Zariski|first= Oscar|title=Sur la normalité analytique des variétés normales|journal=Annales de l'Institut Fourier |volume=2 |year=1950|pages= 161–164|doi=10.5802/aif.27|url=http://www.numdam.org/item?id=AIF_1950__2__161_0|doi-access=free}} *{{Citation | last1=Zariski | first1=Oscar | author1-link=Oscar Zariski | last2=Samuel | first2=Pierre | author2-link=Pierre Samuel | title=Commutative algebra. Vol. II | orig-date=1960 | publisher=Springer-Verlag | location=Berlin, New York | isbn=978-0-387-90171-8 |mr=0389876 | year=1975}}
Category:Commutative algebra
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