Editing Commutative algebra (section)

===Zariski topology on prime ideals===
{{Main|Zariski topology}}
The [[Zariski topology]] defines a [[topological space|topology]] on the [[spectrum of a ring]] (the set of prime ideals).<ref>{{cite book
| last1 = Dummit
| first1 = D. S.
| last2 = Foote
| first2 = R.
| title = Abstract Algebra
| url = https://archive.org/details/abstractalgebra00dumm_304
| url-access = limited
| publisher = Wiley
| pages = [https://archive.org/details/abstractalgebra00dumm_304/page/n84 71]–72
| year = 2004
| edition = 3
| isbn = 9780471433347 
}}</ref>  In this formulation, the Zariski-closed sets are taken to be the sets

:<math>V(I) = \{P \in \operatorname{Spec}\,(A) \mid I \subseteq P\}</math>

where ''A'' is a fixed commutative ring and ''I'' is an ideal.  This is defined in analogy with the classical Zariski topology, where closed sets in affine space are those defined by polynomial equations . To see the connection with the classical picture, note that for any set ''S'' of polynomials (over an algebraically closed field), it follows from [[Hilbert's Nullstellensatz]] that the points of ''V''(''S'') (in the old sense) are exactly the tuples (''a<sub>1</sub>'', ..., ''a<sub>n</sub>'') such that the ideal (''x<sub>1</sub>'' - ''a<sub>1</sub>'', ..., ''x<sub>n</sub>'' - ''a<sub>n</sub>'') contains ''S''; moreover, these are maximal ideals and by the "weak" Nullstellensatz, an ideal of any affine coordinate ring is maximal if and only if it is of this form.  Thus, ''V''(''S'') is "the same as" the maximal ideals containing ''S''.  Grothendieck's innovation in defining Spec was to replace maximal ideals with all prime ideals; in this formulation it is natural to simply generalize this observation to the definition of a closed set in the spectrum of a ring.