Prime ideal

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 important property of unique factorisation expressed in the fundamental theorem of arithmetic does not hold in every ring of algebraic integers, but a substitute was found when Richard Dedekind replaced elements by ideals and prime elements by prime ideals; see Dedekind domain.

The notion of a prime ideal can be generalized to noncommutative rings by using the commutative definition "ideal-wise".

Equivalent formulations of the ideal P ≠ R being prime include the following properties: Prime ideals in commutative rings are characterized by having multiplicatively closed complements in R, and with slight modification, a similar characterization can be formulated for prime ideals in noncommutative rings.

A nonempty subset S ⊆ R is called an m-system if for any a and b in S, there exists r in R such that arb is in S.[8] The following item can then be added to the list of equivalent conditions above: Prime ideals can frequently be produced as maximal elements of certain collections of ideals.