1.
An internal operation on a set X
is a mapping from
2.
An external operation on a set X
is a mapping from
A group is a set X together with an internal operation
such that
a)
the operation is associative, i.e.,
b)
there exists an identity
c)
there is an inverse x^{ –1} for every
A ring
is a set X together with 2 internal
operations
a) X is an abelian group under addition.
b)
Multiplication is, for all
i.
associative:
ii. distributive with respect to addition:
If, in addition, there is an element
X is called a ring with identity (unity).
An element
A ring with identity is called a field if all its elements except zero (neutral element of addition) are regular.
A module X over the ring R is an abelian group X together with an external operation, called scalar multiplication,
such that for all
Furthermore, if the ring R has an identity e, then
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
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.
A linear
(vector) space X is a module in
which the ring of operators is a field. Usually, the field is
N_{Op} \ N_{Set} 
1 
2 
1 
Group (X ; *) 

2 
Ring ( X ; +, * ) Field ( X ; +, * ) 
Module ( X, R ; +, a ) Linear Space (X, K ; +, a ) 
3 

Algebra ( X, R ; +, *, a ) 