The set MaxSpec(R) of maximal ideals of a commutative ring, with the induced Zariski topology.
Let R be a commutative ring. A maximal ideal of R is a proper ideal m⊊R such that there is no ideal strictly between m and R; equivalently, R/m is a field.
The maximal spectrum of R is the set
MaxSpec(R):={m⊂R∣m is a maximal ideal}.
There is always an inclusion MaxSpec(R)⊆Spec(R) (see prime spectrum), since every maximal ideal is prime. One typically topologizes MaxSpec(R) by the subspace topology induced from the Zariski topology on \operatorname{Spec}(R). Concretely, for an ideal I⊆R the corresponding closed subset of MaxSpec(R) is
V(I)∩MaxSpec(R)={m∈MaxSpec(R)∣I⊆m}.
A point m∈MaxSpec(R) has residue field R/m, which agrees with the residue field at m.
The integers. For R=Z, the maximal ideals are exactly (p) for primes p. Thus
MaxSpec(Z)={(p)∣pprime},
which is Spec(Z) with the generic point (0) removed.
Polynomial rings over an algebraically closed field. If k is algebraically closed and R=k[x1,…,xn], then the Nullstellensatz identifies maximal ideals with points a=(a1,…,an)∈kn via
a⟷(x1−a1,…,xn−an).
In this sense, MaxSpec(k[x1,…,xn]) recovers affine n-space over k as a set, and its induced topology is the classical Zariski topology on kn.
A commutative ring is a ringR such that ab=ba for all a,b∈R.
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)∣pprime}.
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.
The Zariski topology on Spec(R) is the topology for which the sets V(I) are precisely the closed subsets, i.e. a subset Z⊆Spec(R) is closed if and only if Z=V(I) for some ideal I.
A commutative ringR is a local ring if it has a unique maximal ideal. One often records this ideal and writes (R,m), where m is the unique maximal ideal.
For a commutative ring R, the following are equivalent:
R is local (i.e. it has a unique maximal ideal).
The set of nonunits in R is an ideal; this ideal is then the unique maximal ideal.
Whenever a+b=1 in R, at least one of a or b is a unit.
Local rings arise systematically from localization: if p is a prime ideal of R, then localizing at the prime produces the local ring Rp.
Many foundational results in commutative algebra are naturally stated for local rings; for instance, Nakayama's lemma is formulated for finitely generated modules over a local ring.
Examples
Fields. Any fieldk is local: its only maximal ideal is (0).
Localizing Z at a prime. For a prime number p, the ring Z(p) from localization at (p) is local, with maximal ideal pZ(p).
Localizing a polynomial ring at a maximal ideal. If k is a field, then k[x](x) is local with maximal ideal generated by x. More generally, k[x,y](x,y) is local with maximal ideal (x,y).
Let k be an algebraically closed field and let A=k[x1,…,xn]. For an ideal I⊆A, define its zero set
V(I)={a∈kn:f(a)=0 for all f∈I}.
For a subset X⊆kn, define the ideal of functions vanishing on X by
I(X)={f∈A:f(a)=0 for all a∈X}.
The sets V(I) are exactly the closed sets of the Zariski topology on kn.
Theorem (Nullstellensatz, variety–ideal correspondence). With k and A as above, the following hold:
For every ideal I⊆A, one has
I(V(I))=I,
the radical of I.
For every Zariski-closed set X⊆kn, one has
V(I(X))=X.
Consequently, the assignments I↦V(I) and X↦I(X) restrict to mutually inverse, inclusion-reversing bijections between:
radical ideals of A, and
Zariski-closed subsets of kn.
A useful special case is the "weak" form: maximal ideals of A are exactly the ideals (x1−a1,…,xn−an) for points a=(a1,…,an)∈kn. In other words, the maximal spectrumMaxSpec(A) can be identified with kn, and the corresponding residue field at such a maximal ideal is (canonically) k.
Examples
A non-radical ideal with the same zero set as its radical. In k[x], let I=(x2). Then V(I)={0}, but
I(V(I))=(x)=(x2).
This illustrates that taking V(−) forgets nilpotent structure, and the theorem recovers exactly the radical.
A point. In k[x,y], let I=(x,y). Then V(I)={(0,0)}. Conversely, for the closed set X={(0,0)} one has I(X)=(x,y), a maximal ideal, matching the identification of maximal ideals with points.
A reducible algebraic set. In k[x,y], let I=(xy). Then V(I) is the union of the two coordinate axes. The ideal of this union is
I(V(I))=(x)∩(y)=(xy),
reflecting that V(I) has two irreducible components.