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Prime ideal
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===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 of a ring|spectrum]], into a [[topological space]] and can thus define generalizations of varieties called [[scheme (mathematics)|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 integer]]s, but a substitute was found when [[Richard Dedekind]] replaced elements by ideals and prime elements by prime ideals; see [[Dedekind domain]].
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