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.
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<sub>n</sub> and localized at these elements, where
Then R[X]/(X<sup> 2</sup>âÂÂz<sub>1</sub>) is a normal Noetherian local ring that is analytically reducible.