OpenMath Content Dictionary: ringname1

Canonical URL:
http://www.openmath.org/cd/ringname1.ocd
CD Base:
http://www.openmath.org/cd
CD File:
ringname1.ocd
CD as XML Encoded OpenMath:
ringname1.omcd
Defines:
Z, Zm, quaternions
Date:
2004-03-08
Version:
1
Review Date:
2005-04-01
Status:
experimental

A CD of names of frequently used rings in ring theory.

Written by Arjeh M. Cohen 2004-03-08

Z

Description:

This symbol represents the ring of integers.

Commented Mathematical property (CMP):
The integer 1 is the identity element of this ring.
Formal Mathematical property (FMP):
identity ( Z ) = 1
Commented Mathematical property (CMP):
The carrier set of this ring is the set of integers.
Formal Mathematical property (FMP):
carrier ( Z ) = Z
Signatures:
sts


[Next: quaternions] [Last: Zm] [Top]

quaternions

Description:

This symbol represents a unary function. Its argument is a ring R. When evaluated on R, the function represents the ring of quaternions over R, that is, the ring with basis 1,i,j,k over R such that ij=-ji=k, i^2=j^2=k^2=-1.

Commented Mathematical property (CMP):
The quaternion ring over R is isomorphic to the quotient of the free ring over R generated by i, j, k subject to the relations ij=-ji=k and i^2=j^2=k^2=-1.
Formal Mathematical property (FMP):
isomorphic ( quaternions ( R ) , quotient_ring ( free_ring ( Q , i , j , k ) , ideal ( free_ring ( Q , i , j , k ) , i j - k j i + k i 2 + 1 j 2 + 1 k 2 + 1 ) ) )
Signatures:
sts


[Next: Zm] [Previous: Z] [Top]

Zm

Role:
application
Description:

This symbol represents the ring of integers modulo m, where m is not necessarily a prime. It takes one argument, the integer m.

Example:
The ring of integers mod 12:
Zm ( 12 )
Signatures:
sts


[First: Z] [Previous: quaternions] [Top]