# Notes on Abstract Algebra

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

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:

- For all
*a*,*b*and*c*in*S*, (*a***b*) **c*=*a** (*b***c*)

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

- There is an element
*e*in*M*such that for all*a*in*M*,*a***e*=*e***a*=*a*

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:

- For all
*a*in*G*, there is an element*b*in*G*such that*a***b*=*b***a*=*e*, where*e*is the identity element from the monoid 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:

- (
*R*, +) is an abelian group. The identity element is called 0. - (
*R*, *) is a monoid. The identity element is called 1. - Multiplication distributes over addition, which is to say:
*a** (*b*+*c*) = (*a***b*) + (*a***c*)- (
*a*+*b*) **c*= (*a***c*) + (*b***c*)

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:

- Vector addition:
*V*×*V*→*V*denoted**v**+**w**where**v**,**w**∈*V* - Scalar multiplication:
*F*×*V*→*V*denoted*a***v**where*a*∈*F*,**v**∈*V*

which satisfy these axioms:

*V*under vector addition is an abelian group- Scalar multiplication is associative:
*a*(*b***v**) = (*ab*)**v** - 1
**v**=**v**, where 1 is the multiplicative identity in*F* - Scalar multiplication distributes over vector addition:
*a*(**v**+**w**) =*a***v**+*a***w** - Scalar multiplication distributes over scalar addition: (
*a*+*b*)**v**=*a***v**+*b***v**

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