to parameterize the family of lines through the origin of over by letting and 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 algebraically closed (say, the complex numbers), every maximal ideal corresponds to a point in n-space, by the nullstellensatz (the maximal ideal generated by corresponds to the point ). These representations of are then parametrized by the dual space the covector being given by sending each to the corresponding . Thus a representation of (K-linear maps ) is 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. Functional analysis perspective The term "spectrum" comes from the use in operator theory. Given a linear operator T on a finite-dimensional vector space V, one can consider the vector space with operator as a module over the polynomial ring in one variable R=K[T], as in the structure theorem for finitely generated modules over a principal ideal domain. Then the spectrum of K[T] (as a ring) equals the spectrum of T (as an operator). Further, the geometric structure of the spectrum of the ring (equivalently, the algebraic structure of the module) captures the behavior of the spectrum of the operator, such as algebraic multiplicity and geometric multiplicity. For instance, for the 2×2 identity matrix has corresponding module: the 2×2 zero matrix has