OpenMath Content Dictionary: ring3

Canonical URL:
http://www.openmath.org/cd/ring3.ocd
CD File:
ring3.ocd
CD as XML Encoded OpenMath:
ring3.omcd
Defines:
direct_power, direct_product, free_ring, ideal, integers, invertibles, is_ideal, kernel, m_poly_ring, matrix_ring, multiplicative_group, poly_ring, principal_ideal, quotient_ring
Date:
2004-06-01
Version:
1 (Revision 1)
Review Date:
2006-06-01
Status:
experimental

A CD of functions for basic constructions in ring theory. The quaternion definition is still very shaky.

Written by Arjeh M. Cohen 2004-02-25

is_ideal

Description:

The binary boolean function whose value is true if and only if the second argument is an ideal of the second.

Commented Mathematical property (CMP):
If is_ideal(S,I) then I is a nonempty set of elements of S and I is a subgroup of the additive group of S and closed under multiplication by elements of S.
Signatures:
sts


[Next: ideal] [Last: integers] [Top]

ideal

Description:

This symbol represents a binary function. The first argument is a ring R and the second argument is a list or a set. When evaluated on R and such a second argument, the function represents the ideal in R generated by the entries of the list or set.

Example:
The ideal in the free ring on the letters a, b generated by a*b-b*a:
  
ideal ( free_ring ( ( a , b ) ) , a b - b a )
Signatures:
sts


[Next: kernel] [Previous: is_ideal] [Top]

kernel

Description:

This symbol represents a unary function. Its argument is a ring homomorphism f : R -> S. When evaluated on f, the function represents the kernel in R of f, that is, the subset {x in R | f(x) = 0}.

Commented Mathematical property (CMP):
The kernel of a ring homomorphism is an ideal.
Formal Mathematical property (FMP):
  
is_homomorphism ( R , S , f ) is_ideal ( R , kernel ( f ) )
Signatures:
sts


[Next: principal_ideal] [Previous: ideal] [Top]

principal_ideal

Description:

This symbol represents a binary function. The first argument is a ring R and the second argument is an element of R. When evaluated on R and such a second argument, the function represents the ideal in R generated by the second argument.

Example:
The ideal in the free ring over the rationals on the letters a, b generated by a*b-b*a:
  
principal_ideal ( free_ring ( Q , ( a , b ) ) , a b - b a )
Signatures:
sts


[Next: free_ring] [Previous: kernel] [Top]

free_ring

Description:

This symbol represents a binary function. The first argument should be a ring and the second a list or a set. When evaluated on such arguments R and L, the function represents the free ring over R generated by the elements (or entries) of L. This ring can also be viewed as the ring of non-commutative polynomials over R with variables the elements of L.

Example:
The free ring over R on the letters a, b:
  
free_ring ( R , ( a , b ) )
Signatures:
sts


[Next: poly_ring] [Previous: principal_ideal] [Top]

poly_ring

Description:

This symbol represents a binary function. The first argument should be a ring and the second a variable. When evaluated on such arguments R and X, the function represents the free commutative ring over R generated by X. This ring can also be viewed as the ring of polynomials over R with indeterminate X.

Example:
The polynomial ring over R with indeterminate X:
  
poly_ring ( R , X )
Signatures:
sts


[Next: m_poly_ring] [Previous: free_ring] [Top]

m_poly_ring

Description:

This symbol represents a binary function. The first argument should be a ring and the second a list or a set. When evaluated on such arguments R and L, the function represents the free commutative ring over R generated by the elements (or entries) of L. This ring can also be viewed as the ring of polynomials over R with variables the elements of L.

Example:
The polynomial ring over R with variables a, b:
  
m_poly_ring ( R , ( a , b ) )
Signatures:
sts


[Next: matrix_ring] [Previous: poly_ring] [Top]

matrix_ring

Description:

This symbol represents a binary function. The first argument is a positive integer n, the second is a ring R. When evaluated on such argument n and R, the function represents the ring of n x n matrices over R.

Commented Mathematical property (CMP):
The ring of 1 x 1 matrices over R is isomorphic to R.
Formal Mathematical property (FMP):
  
isomorphic ( matrix_ring ( 1 , R ) , R )
Signatures:
sts


[Next: direct_product] [Previous: m_poly_ring] [Top]

direct_product

Description:

This is a symbol with two or more arguments, all of which are rings. It denotes the ring that is the direct product of its arguments.

Signatures:
sts


[Next: direct_power] [Previous: matrix_ring] [Top]

direct_power

Description:

This is a symbol with two arguments. The first argument should be a ring S and the second argument a positive integer n. It denotes the direct product of n copies of S.

Example:
  
direct_product ( ring ( Z , + , - , 0 ) , ring ( Z , + , - , 0 ) ) = direct_power ( ring ( Z , + , - , 0 ) , 2 )
Signatures:
sts


[Next: quotient_ring] [Previous: direct_product] [Top]

quotient_ring

Description:

This is a binary function, whose first argument is a ring R and whose second argument is an ideal I of R. When applied to R and I, it denotes the quotient ring of R by I.

Example:
The carrier of the ring of integers modulo 2 is introduced as Zm(2) in the CD setname2. The ring can also be defined as follows.
  
quotient_ring ( ring ( Z , + , 0 , - , × , 1 ) , ideal ( ring ( Z , + , 0 , - , × , 1 ) , ( 2 ) ) )
Example:
The ring (Z/2Z)[x]/(x^2+x+1)
  
quotient_ring ( poly_ring ( Z 2 , x ) , ideal ( poly_ring ( Z 2 , x ) , x 2 + x + 1 ) )
Using the xref mechanism it can also be represented as
  
quotient_ring ( poly_ring ( Z 2 , x ) , principal_ideal ( poly_ring ( Z 2 , x ) , x 2 + x + 1 ) )
Signatures:
sts


[Next: multiplicative_group] [Previous: direct_power] [Top]

multiplicative_group

Description:

This is a unary function, whose argument is a ring R. When applied to R, it denotes the group of invertible elements of R with respect to the multiplication on R.

Commented Mathematical property (CMP):
The multiplicative group of the ring R is the group of invertible elements of the multiplicative monoid of R.
Formal Mathematical property (FMP):
  
invertibles ( R ) = invertibles ( multiplicative_monoid ( R ) )
Signatures:
sts


[Next: invertibles] [Previous: quotient_ring] [Top]

invertibles

Description:

This is a unary function, whose argument is a ring R. When applied to R, it denotes the set of invertible elements of R with respect to the multiplication on R.

Commented Mathematical property (CMP):
The carrier of the multiplicative group of the ring R is the set of invertible elements of R.
Formal Mathematical property (FMP):
  
invertibles ( R ) = carrier ( multiplicative_group ( R ) )
Signatures:
sts


[Next: integers] [Previous: multiplicative_group] [Top]

integers

Description:

This is a symbol representing the ring of integers.

Commented Mathematical property (CMP):
The ring of integers is (Z, +,0,-,*,1), where +,-,* are the standard arithmetic operations.
Formal Mathematical property (FMP):
  
integers = ring ( Z )
Signatures:
sts


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