Let R be a commutative ring. A prime ideal of R is a proper ideal p⊊R such that whenever ab∈p (with a,b∈R), then a∈p or b∈p.
The prime spectrum of R is the set
Spec(R):={p⊂R∣p is a prime ideal}.
An element p∈Spec(R) is called a point of Spec(R).
In commutative algebra one usually studies Spec(R) together with the Zariski topology; this turns Spec(R) into a topological space whose basic opens are closely related to localizations. For a point p∈Spec(R), the associated local data are the localization Rp and its residue field κ(p).
Examples
- A field has a one-point spectrum. If k is a field, the only prime ideal is (0), so Spec(k)={(0)}.
- The spectrum of the integers. In R=Z, the prime ideals are (0) and (p) for primes p. Thus
Spec(Z)={(0)} ∪ {(p)∣p prime}. Under the Zariski topology, the point (0) is a generic point whose closure is all of Spec(Z).
- The spectrum of a polynomial ring in one variable. Let k be a field and R=k[x]. Then (0) is prime, and every nonzero prime ideal is generated by an irreducible polynomial. So
Spec(k[x])={(0)} ∪ {(f)∣f∈k[x] irreducible}. If k is algebraically closed, the maximal ideals are precisely (x−a), and MaxSpec(k[x]) can be identified with the affine line over k.