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 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 is the study of modules over its group algebra.
The connection to representation theory is clearer if one considers the polynomial ring or, without a basis, As the latter formulation makes clear, a polynomial ring is the group algebra over a vector space, and writing in terms of corresponds to choosing a basis for the vector space. Then an ideal I, or equivalently a module is a cyclic representation of R (cyclic meaning generated by 1 element as an R-module; this generalizes 1-dimensional representations).
In the case that the field is closed (say, the complex numbers) and one uses a maximal ideal, which corresponds (by the nullstellensatz) to a point in n-space (the maximal ideal generated by corresponds to the point ), these representations are parametrized by the dual space (the covector is given by the ). This is precisely Fourier theory: the representations the additive group are given by the dual group (simply, maps are multiplication by a scalar), and thus the representations of (K-linear maps ) are given by a set of n-numbers, or equivalently a covector
Thus, points in n-space, thought of as the max spec of 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.
Read more about this topic: Spectrum Of A Ring
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