Written on: 2003-09-21

Notes on Abstract Algebra

A binary operation on a set S is a binary function : S × SS.

A magma (M, *) is a set M together with a binary operation * on M.

A semigroup (S, *) is an associative magma, that is to say that it is a magma that satisfies this axiom:

A monoid (M, *) is a unital semigroup, that is to say that it is a semigroup that satisfies this additional axiom:

The element e in the above axiom is the identity element.

A group (G, *) is a monoid whose elements are invertible, that is to say that it is a monoid that satisfies this additional axiom:

If for all a and b in G, a * b = b * a then the group is commutative and is called an abelian group.

A ring (R, +, *) is a set R with two binary operations + and * (commonly referred to as addition and multiplication) such that:

If multiplication is commutative, then it is a commutative ring.

A field is a commutative ring such that 0 does not equal 1 and every element except 0 has a multiplicative inverse.

A vector space over a field F is a set V together with two operations:

which satisfy these axioms:

The elements of V are called vectors and the elements of F are called scalars.