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.
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. Nagata (1958, 1962, Appendix A1, example 7) 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 [[x,y]] is the formal power series ring over K in 2 variables. Let R be the subring of K [[x,y]] generated by x, y, and the elements zn and localized at these elements, where
Then R[X]/(X 2–z1) is a normal Noetherian local ring that is analytically reducible.