# Algebraic Structures

### Operations

1.         An internal operation on a set X is a mapping from X×X  into X.

2.         An external operation on a set X is a mapping from A×X  into X.  Here, A is another set whose elements are called operators on X.

### Groups

A group is a set X together with an internal operation

X×X    X   by  ( x,y )    xy

such that

a)        the operation is associative, i.e., ( xy )z=x( yz )  for all x,y,zX .

b)        there exists an identity eX  such that xe=ex=x  for all xX .

c)        there is an inverse x –1 for every xX  such that x 1 x=x x 1 =e .

### Rings

A ring is a set X together with 2 internal operations ( x,y )    xy  and ( x,y )    x+y  called multiplication and addition, respectively, such that

a)        X is an abelian group under addition.

b)        Multiplication is, for all x,y,zX ,

i.              associative:   ( xy )z=x( yz ) .

ii.              distributive with respect to addition:

x( y+z )=xy+xz   and  ( y+z )x=yx+zx .

If, in addition, there is an element eX  such that

ex=xe=x         for all xX

X is called a ring with identity (unity).

An element xX  that has an inverse x 1  is called regular (invertible, non-singular).

### Fields

A ring with identity is called a field if all its elements except zero (neutral element of addition) are regular.

### Modules

A module X over the ring R is an abelian group X together with an external operation, called scalar multiplication,

R×X    X  by ( α,x )    αx

such that for all α,βR  and x,yX , we have

α( x+y )=αx+αy .

( α+β )x=αx+βx

( αβ )x=α( βx )

Furthermore, if the ring R has an identity e, then

ex=x

### Algebras

An algebra A is a module over a ring R with identity together with an internal associative operation, usually called multiplication, such that

1.         A is a ring.

2.         the external operation ( α,x )    αx  obeys

α( xy )=( αx )y=x( αy )

Most algebras are linear (vector) spaces together with an internal operation called multiplication.  The definition given above is much more general and includes, for example, algebra of tensor fields over the ring of functions.

### Linear Spaces

A linear (vector) space X is a module in which the ring of operators is a field. Usually, the field is 𝕂=  or     .  Elements of X are called vectors.

### Summary

 NOp \ NSet 1 2 1 Group (X ; *) 2 Ring ( X ; +, * ) Field ( X ; +, * ) Module ( X, R ; +, a ) Linear Space (X, K ; +, a ) 3 Algebra ( X, R ; +, *, a )