In algebra, an analytically normal ring is a local ring whose completion is a normal ring, in other words a domain that is integrally closed in its quotient field.
Zariski (1950) proved that if a local ring of an algebraic variety is normal, then it is analytically normal, which is in some sense a variation of Zariski's main theorem.
Nagata (1958, 1962, Appendix A1, example 7) gave an example of a normal Noetherian local ring that is analytically reducible and therefore not analytically normal.
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