The importance of the idea in general of a module, more general than an ideal, probably led to the perception that ideal theory was too narrow a description. Valuation theory, too, was an important technical extension, and was used by Helmut Hasse and Oscar Zariski. Bourbaki used commutative algebra; sometimes local algebra is applied to the theory of local rings. D. G. Northcott's 1953 Cambridge Tract Ideal Theory (reissued 2004 under the same title) was one of the final appearances of the name.
In ring theory, a branch of abstract algebra, an ideal is a special subset of a ring.
Ideals are important because they appear as the kernels of ring homomorphisms and allow one to define factor ring.
The sum and the intersection of ideals is again an ideal; with these two operations as join and meet, the set of all ideals of a given ring forms a lattice.
The existence of prime ideals is in general not obvious, and often a satisfactory amount of prime ideals cannot be derived within Zermelo-Fraenkel set theory.
In Stone's representation theorem for Boolean algebras, the maximal ideals (or, equivalently via the negation map, ultrafilters) are used to obtain the set of points of a topological space, whose clopen sets are isomorphic to the original Boolean algebra.
Ideals were introduced first by Marshall H. Stone, who derived their name from the ring ideals of abstract algebra.
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