TABLE OF CONTENTS In comparison with classical mechanics and field theory phrased in terms of smooth finite-dimensional manifolds, quantum theory speaks the algebraic language adapted to describing systems of infinite degrees of freedom. Geometric techniques are involved in quantum theory due to the fact that the differential calculus over an arbitrary ring can be defined. Their relation to the familiar differential geometry of smooth manifolds is based on the fact that any manifold can be characterized in full by a certain algebraic construction and, furthermore, there is the categorial equivalence between the vector bundles over a smooth manifold and the finite projective modules over the ring of smooth real functions on this manifold.
In this Section, the relevant basics on modules over commutative algebras is summarized [272; 288].
An algebra 
is an additive group which is additionally provided with distributive multiplication. All algebras throughout the book are associative, unless they are Lie algebras. A ring is a unital algebra, i.e., it contains a unit element 1. Unless otherwise stated, we assume that 1 ? 0, i.e., a ring does not reduce to the zero element. One says that A is a division algebra if it has no a divisor of zero, i.e., ab = 0, a, b ? A, implies either a = 0 or b = 0. Non-zero elements of a ring form a multiplicative monoid. If this multiplicative monoid is a multiplicative group, one says that...
