For a domain R, normality means R is integrally closed in its field of fractions; for a scheme, normal means all local rings are integrally closed domains.

Principle

Normality is organized by integrality: elements of the fraction field that satisfy a monic polynomial relation with coefficients in R already lie in R. It controls extension of divisors and codimension-one behavior.

Demonstration

Any unique factorization domain (UFD) is normal; the coordinate ring k[x1,...,xn] and local rings of smooth varieties are normal, whereas a cusp coordinate ring like k[t^2,t^3] is not normal and requires normalization.

Misapplication

Confusing normality with factoriality (UFD) or with regularity; assuming integrally closed implies all desirable factorization properties or smoothness is incorrect.

Consequence

Normal rings permit a well-behaved divisor theory in codimension one, normalization maps are finite under mild hypotheses, and certain local cohomology and extension phenomena simplify on normal schemes.

Reversal

Non-normal rings have integral elements absent from R; their normalization introduces a finite birational extension that resolves some singularities but may create multiple components.

Boundary

Normality is primarily a property of domains (or of each irreducible component); for non-domains one works componentwise or with integral closures in total ring of fractions; characteristic-dependent pathologies can arise.

Semantic Tension

Normal vs integrally closed vs geometrically normal: integrally closed in the fraction field is the algebraic definition, while geometric normality includes base-change stability and separability conditions over nonperfect fields.

Synthesis

Normality is the integrally closed condition for domains: it forbids hidden integral elements in the fraction field, enabling controlled divisor theory and milder codimension-one singularities.