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2 Properties 3 Uses |
One use of prime ideals occurs in algebraic geometry, where varieties are defined as the zero sets of ideals in polynomial rings. It turns out that the irreducible varieties correspond to prime ideals. In the modern abstract approach, one starts with an arbitrary commutative ring and turns the set of its prime ideals, also called its spectrum, into a topological space and can thus define generalizations of varieties called schemes, which find applications not only in geometry, but also in number theory.
The introduction of prime ideals in algebraic number theory was a major step forward: it was realized that the ordinary fundamental theorem of arithmetic does not work in rings of algebraic integers, but a substitute was found when Dedekind replaced elements by ideals and prime elements by prime ideals; see Dedekind domain.Examples
Properties
Uses