Hilbert C*-module
Hilbert C*-modules are mathematical objects which generalise the notion of a Hilbert space (which itself is a generalisation of Euclidean space), in that they endow a linear space with an "inner product" which takes values in a C*-algebra. Hilbert C*-modules were first introduced in the work of Irving Kaplansky in 1953, which developed the theory for commutative, unital algebras (though Kaplansky observed that the assumption of a unit element was not "vital"). In the 1970s the theory was extended to non-commutative C*-algebras independently by William Lindall Paschke and Marc Rieffel, the latter in a paper which used Hilbert C*-modules to construct a theory of induced representations of C*-algebras. Hilbert C*-modules are crucial to Kasparov's formulation of KK-theory, and provide the right framework to extend the notion of Morita equivalence to C*-algebras. They can be viewed as the generalization of vector bundles to noncommutative C*-algebras and as such play an important role in noncommutative geometry, notably in C*-algebraic quantum group theory, and groupoid C*-algebras.
Hilbert's fourteenth problem
gave the following counterexample to Hilbert's problem. The field "k" is a field containing 48 elements "a", ...,"a", for "i"=1, 2, 3 that are algebraically independent over the prime field. The ring "R" is the polynomial ring "k"["x"...,"x", "t"...,"t"] in 32 variables. The vector space "V" is a 13-dimensional vector space over "k" consisting of all vectors ("b"...,"b") in "k" orthogonal to each of the three vectors ("a", ...,"a") for "i"=1, 2, 3. The vector space "V" is a 13-dimensional commutative unipotent algebraic group under addition, and its elements act on "R" by fixing all elements "t" and taking "x" to "x" + "b""t". Then the ring of elements of "R" invariant under the action of the group "V" is not a finitely generated "k"-algebra.