In algebra, an analytically irreducible ring is a

local ring In mathematics, more specifically in ring theory, local rings are certain rings that are comparatively simple, and serve to describe what is called "local behaviour", in the sense of functions defined on algebraic varieties or manifolds, or of ...

whose completion has no zero divisors. Geometrically this corresponds to a variety with only one analytic branch at a point. proved that if a local ring of an algebraic variety is a

normal ring In commutative algebra, an integrally closed domain ''A'' is an integral domain whose integral closure in its field of fractions is ''A'' itself. Spelled out, this means that if ''x'' is an element of the field of fractions of ''A'' that is a roo ...

, 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. gave such an example of a normal Noetherian local ring that is analytically reducible.

Nagata's example

Suppose that ''K'' is a field of characteristic not 2, and ''K'' is the formal

power series In mathematics, a power series (in one variable) is an infinite series of the form \sum_^\infty a_n \left(x - c\right)^n = a_0 + a_1 (x - c) + a_2 (x - c)^2 + \dots where ''a_n'' represents the coefficient of the ''n''th term and ''c'' is a co ...

ring over ''K'' in 2 variables. Let ''R'' be the subring of ''K'' generated by ''x'', ''y'', and the elements ''z''_''n'' and localized at these elements, where :

w=\sum_ a_mx^m

is transcendental over ''K''(''x'') :

z_1=(y+w)^2

z_=(z_1-(y+\sum_a_mx^m)^2)/x^n

. Then ''R'' 'X''(''X''²–''z''₁) is a normal Noetherian local ring that is analytically reducible.

References

* * * * Commutative algebra {{commutative-algebra-stub