A local ring of an algebraic variety or a completion of such a ring. A commutative ring obtained from a ring of polynomials over a field by means of the operations of completion, localization and factorization by a prime ideal is called an algebro-geometric ring . A local ring of an irreducible algebraic variety does not obtain nilpotent elements as a result of completion . This property of a local ring is known as analytic reducibility. A similar fact concerning local rings of normal varieties  is that the completion of a local ring of a normal algebraic variety is a normal ring (analytic normality). Examples of local Noetherian rings that are not analytically reduced or analytically normal are known . A pseudo-geometric ring is a Noetherian ring any quotient ring of which by a prime ideal is a Japanese ring. An integral domain $A$ is called a Japanese ring if its integral closure in a finite extension of the field of fractions is a finite $A$-module . The class of pseudo-geometric rings is closed with respect to localizations and extensions of finite type; it includes the ring of integers and all complete local rings. See also Excellent ring.
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Geometric ring. Encyclopedia of Mathematics. URL: http://www.encyclopediaofmath.org/index.php?title=Geometric_ring&oldid=31943