Some algebraic structures have more than one law of composition. These must be connected by some kind of distributive laws, else the separate laws of composition are simply independent structures on the same set. The most elementary algebraic structures of this kind are known as rings and fields, and by combining fields and abelian groups we create vector spaces [1–7].

For the rest of this book, vector spaces will never be far away. For example, Hilbert spaces are structured vector spaces that form the basis of quantum mechanics. Even in non-linear theories such as classical mechanics and general relativity there exist local vector spaces known as the tangent space at each point, which are needed to formulate the dynamical equations. It is hard to think of a branch of physics that does not use vector spaces in some aspect of its formulation.

Rings and fields

A ringR is a set with two laws of composition called addition and multiplication, denoted a + b and ab respectively. It is required that R is an abelian group with respect to +, with identity element 0 and inverses denoted −a. With respect to multiplication R is to be a commutative semigroup, so that the identity and inverses are not necessarily present. In detail, the requirements of a ring are:

(R1) Addition is associative, (a + b) + c = a + (b + c).

(R2) Addition is commutative, a + b = b + a.

(R3) There is an element 0 such that a + 0 = a for all a ∈ R.