Analytically irreducible ring explained

In algebra, an analytically irreducible ring is a local ring 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, 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 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 _{m>0} a ^{m} _{mx} is transcendental over K(x)z ^{2} _{1= (y + w)} z _{n+1} = (z _{1-(y+\sum} _{0<m<n} a ^{m)} _{mx} ^{2)/x} ^{n} .Then R[''X'']/(X ²–z₁) is a normal Noetherian local ring that is analytically reducible