# OpenMath Content Dictionary: ring3

Canonical URL:
http://www.openmath.org/cd/ring3.ocd
CD Base:
http://www.openmath.org/cd
CD File:
ring3.ocd
CD as XML Encoded OpenMath:
ring3.omcd
Defines:
ideal, kernel, principal_ideal, direct_power, direct_product, free_ring, integers, invertibles, is_ideal, m_poly_ring, matrix_ring, multiplicative_group, poly_ring, 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:
$\mathrm{ideal}\left(\mathrm{free_ring}\left(\left(a,b\right)\right),\begin{array}{c}ab-ba\hfill \end{array}\right)$
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):
$\mathrm{is_homomorphism}\left(R,S,f\right)⇒\mathrm{is_ideal}\left(R,\mathrm{kernel}\left(f\right)\right)$
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:
$\mathrm{principal_ideal}\left(\mathrm{free_ring}\left(Q,\left(a,b\right)\right),ab-ba\right)$
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:
$\mathrm{free_ring}\left(R,\left(a,b\right)\right)$
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:
$\mathrm{poly_ring}\left(R,X\right)$
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:
$\mathrm{m_poly_ring}\left(R,\left(a,b\right)\right)$
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):
$\mathrm{isomorphic}\left(\mathrm{matrix_ring}\left(1,R\right),R\right)$
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:
$\mathrm{direct_product}\left(\mathrm{ring}\left(\mathbb{Z},+,-,0\right),\mathrm{ring}\left(\mathbb{Z},+,-,0\right)\right)=\mathrm{direct_power}\left(\mathrm{ring}\left(\mathbb{Z},+,-,0\right),2\right)$
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.
$\mathrm{quotient_ring}\left(\mathrm{ring}\left(\mathbb{Z},+,0,-,×,1\right),\mathrm{ideal}\left(\mathrm{ring}\left(\mathbb{Z},+,0,-,×,1\right),\left(2\right)\right)\right)$
Example:
The ring (Z/2Z)[x]/(x^2+x+1)
$\mathrm{quotient_ring}\left(\mathrm{poly_ring}\left({\mathbb{Z}}_{2},x\right),\mathrm{ideal}\left(\mathrm{poly_ring}\left({\mathbb{Z}}_{2},x\right),\begin{array}{c}{x}^{2}+x+1\hfill \end{array}\right)\right)$
Using the xref mechanism it can also be represented as
$\mathrm{quotient_ring}\left(\mathrm{poly_ring}\left({\mathbb{Z}}_{2},x\right),\mathrm{principal_ideal}\left(\mathrm{poly_ring}\left({\mathbb{Z}}_{2},x\right),{x}^{2}+x+1\right)\right)$
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):
$\mathrm{invertibles}\left(R\right)=\mathrm{invertibles}\left(\mathrm{multiplicative_monoid}\left(R\right)\right)$
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):
$\mathrm{invertibles}\left(R\right)=\mathrm{carrier}\left(\mathrm{multiplicative_group}\left(R\right)\right)$
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):
$\mathrm{integers}=\mathrm{ring}\left(\mathbb{Z}\right)$
Signatures:
sts

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