Download Algebraic Structures

Algebraic Structures
Operations
1.
2.
An internal operation on a set X is a mapping from X  X into X.
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
 xy  z  x  yz 
for all x, y, z  X .
a)
the operation is associative, i.e.,
b)
there exists an identity e  X such that xe  ex  x for all x  X .
c)
there is an inverse x –1 for every x  X such that x 1 x  xx 1  e .
Rings
A ring is a set X together with 2 internal operations
 x, y 
xy and
 x, y 
called multiplication and addition, respectively, such that
a) X is an abelian group under addition.
b) Multiplication is, for all x, y, z  X ,
 xy  z  x  yz  .
i.
associative:
ii.
distributive with respect to addition:
x  y  z   xy  xz
and
 y  z  x  yx  zx .
If, in addition, there is an element e  X such that
for all x  X
ex  xe  x
X is called a ring with identity (unity).
An element x  X that has an inverse x 1 is called regular (invertible,
non-singular).
x y
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  , x 
x
such that for all  ,   R and x, y  X , we have
  x  y  x  y .
    x   x   x
  x     x 
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  , x 
 x obeys
  xy    x  y  x  y 
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  R or C . Elements of X are called vectors.
Summary
NOp \ NSet
1
2
1
2
3
Group (X ; *)
Ring ( X ; +, * )
Module ( X, R ; +,  )
Field ( X ; +, * ) Linear Space (X,  ; +,  )
Algebra ( X, R ; +, *,  )