Hilbert space
The algebra of observables in quantum mechanics is naturally an algebra of operators defined on a Hilbert space, according to Werner Heisenberg's matrix mechanics formulation of quantum theory. Von Neumann began investigating operator algebras in the 1930s, as rings of operators on a Hilbert space. The kind of algebras studied by von Neumann and his contemporaries are now known as von Neumann algebras. In the 1940s, Israel Gelfand, Mark Naimark and Irving Segal gave a definition of a kind of operator algebras called C*-algebras that on the one hand made no reference to an underlying Hilbert space, and on the other extrapolated many of the useful features of the operator algebras that had previously been studied. The spectral theorem for self-adjoint operators in particular that underlies much of the existing Hilbert space theory was generalized to C*-algebras. These techniques are now basic in abstract harmonic analysis and representation theory.
Hilbert series and Hilbert polynomial
The ring formula_51 has Krull dimension one, and is the ring of regular functions of a projective algebraic set formula_52 of dimension 0 consisting of a finite number of points, which may be multiple points. As formula_53 belongs to a regular sequence, none of these points belong to the hyperplane of equation formula_54 The complement of this hyperplane is an affine space that contains formula_55 This makes formula_52 an affine algebraic set, which has formula_57 as its ring of regular functions. The linear polynomial formula_58 is not a zero divisor in formula_59 and one has thus an exact sequence
which implies that 
Here we are using Hilbert series of filtered algebras, and the fact that the Hilbert series of a graded algebra is also its Hilbert series as filtered algebra.