Editing Spectrum of a ring (section)

== Representation theory perspective ==
From the perspective of [[representation theory]], a prime ideal ''I'' corresponds to a module ''R''/''I'', and the spectrum of a ring corresponds to [[Irreducible representation|irreducible]] cyclic representations of ''R'', while more general subvarieties correspond to possibly reducible representations that need not be cyclic. Recall that abstractly, the representation theory of a [[group (mathematics)|group]] is the study of modules over its [[group ring|group algebra]].

The connection to representation theory is clearer if one considers the [[polynomial ring]] <math>R=K[x_1,\dots,x_n]</math> or, without a basis, <math>R=K[V].</math> As the latter formulation makes clear, a polynomial ring is the group algebra over a [[vector space]], and writing in terms of <math>x_i</math> corresponds to choosing a basis for the vector space. Then an ideal ''I,'' or equivalently a module <math>R/I,</math> is a cyclic representation of ''R'' ([[Cyclic module|cyclic]] meaning generated by 1 element as an ''R''-module; this generalizes 1-dimensional representations).

In the case that the field is algebraically closed (say, the complex numbers), every maximal ideal corresponds to a point in ''n''-space, by the [[Nullstellensatz]] (the maximal ideal generated by <math>(x_1-a_1), (x_2-a_2),\ldots,(x_n-a_n)</math> corresponds to the point <math>(a_1,\ldots,a_n)</math>). These representations of <math>K[V]</math> are then parametrized by the [[dual space]] <math>V^*,</math> the covector being given by sending each <math>x_i</math> to the corresponding <math>a_i</math>. Thus a representation of <math>K^n</math> (''K''-linear maps <math>K^n \to K</math>) is given by a set of ''n'' numbers, or equivalently a covector <math>K^n \to K.</math>

Thus, points in ''n''-space, thought of as the max spec of <math>R=K[x_1,\dots,x_n],</math> correspond precisely to 1-dimensional representations of ''R'', while finite sets of points correspond to finite-dimensional representations (which are reducible, corresponding geometrically to being a union, and algebraically to not being a prime ideal). The non-maximal ideals then correspond to ''infinite''-dimensional representations.