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In algebra, an '''analytically irreducible ring''' is a [[local ring]] whose [[completion (ring theory)|completion]] has no zero divisors. Geometrically this corresponds to a variety with only one analytic branch at a point. 

{{harvtxt|Zariski|1948}} proved that if a local ring of an algebraic variety is a [[normal ring]], then it is analytically irreducible. There are many examples of reduced and irreducible local rings that are analytically reducible, such as the local ring of a node of an irreducible curve, but it is hard to find examples that are also normal. {{harvs|txt|last=Nagata|year1=1958|year2=1962|loc2=Appendix A1, example 7}} gave such an example of a normal [[Noetherian ring|Noetherian local ring]] that is analytically reducible.

==Nagata's example==

Suppose that ''K'' is a field of characteristic not 2, and ''K''&nbsp;[<span/>[''x'',''y'']] is the formal power series ring over ''K'' in 2 variables. Let ''R'' be the subring of ''K''&nbsp;[<span/>[''x'',''y'']] generated by ''x'', ''y'', and the elements ''z''<sub>''n''</sub> and localized at these elements, where 
:<math>w=\sum_{m>0} a_mx^m</math> is transcendental over ''K''(''x'')
:<math>z_1=(y+w)^2</math>
:<math>z_{n+1}=(z_1-(y+\sum_{0<m<n}a_mx^m)^2)/x^n</math>.
Then ''R''[''X'']/(''X''<sup> 2</sup>–''z''<sub>1</sub>) is a normal Noetherian local ring that is analytically reducible.

== 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}}
*{{citation| mr=0024158 |last=Zariski|first= Oscar|authorlink=Oscar Zariski|title=Analytical irreducibility of normal varieties|journal=Ann. of Math. (2) |volume=49|year=1948|pages= 352–361|doi=10.2307/1969284}} 
*{{Citation | last1=Zariski | first1=Oscar | author1-link=Oscar Zariski | last2=Samuel | first2=Pierre | author2-link=Pierre Samuel | title=Commutative algebra. Vol. II | origyear=1960 | publisher=[[Springer-Verlag]] | location=Berlin, New York | isbn=978-0-387-90171-8 | id={{MathSciNet | id = 0389876}} | year=1975}}

[[Category:Commutative algebra]]