module showing geometric multiplicity 2 for the zero eigenvalue, while a non-trivial 2×2 nilpotent matrix has module showing algebraic multiplicity 2 but geometric multiplicity 1. In more detail: the eigenvalues (with geometric multiplicity) of the operator correspond to the (reduced) points of the variety, with multiplicity; the primary decomposition of the module corresponds to the unreduced points of the variety; a diagonalizable (semisimple) operator corresponds to a reduced variety; a cyclic module (one generator) corresponds to the operator having a cyclic vector (a vector whose orbit under T spans the space); the last invariant factor of the module equals the minimal polynomial of the operator, and the product of the invariant factors equals the characteristic polynomial. Generalizations The spectrum can be generalized from rings to C*-algebras in operator theory, yielding the notion of the spectrum of a C*-algebra. Notably, for a Hausdorff space, the algebra of scalars (the bounded continuous functions on the space, being analogous to regular functions) is a commutative C*-algebra, with the space being recovered as a topological space from of the algebra of scalars, indeed functorially so; this is the content of the Banach–Stone theorem. Indeed, any commutative C*-algebra can be realized as the algebra of scalars of a Hausdorff space in this way, yielding the same correspondence as between a ring and its spectrum. Generalizing to non-commutative C*-algebras yields noncommutative topology. See also Scheme (mathematics) Projective scheme Spectrum of a matrix Serre's theorem on affineness Étale spectrum Ziegler spectrum Primitive spectrum Citations References External links Kevin R. Coombes: The Spectrum of a Ring Commutative algebra Scheme theory Prime ideals Functional analysis