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):
$\mathrm{identity}\left(Z\right)=1$
Commented Mathematical property (CMP):
The carrier set of this ring is the set of integers.
Formal Mathematical property (FMP):
$\mathrm{carrier}\left(Z\right)=\mathbb{Z}$
Signatures:
sts

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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):
$\mathrm{isomorphic}\left(\mathrm{quaternions}\left(R\right),\mathrm{quotient_ring}\left(\mathrm{free_ring}\left(Q,i,j,k\right),\mathrm{ideal}\left(\mathrm{free_ring}\left(Q,i,j,k\right),\begin{array}{c}ij-k\hfill \\ ji+k\hfill \\ {i}^{2}+1\hfill \\ {j}^{2}+1\hfill \\ {k}^{2}+1\hfill \end{array}\right)\right)\right)$
Signatures:
sts

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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:
$\mathrm{Zm}\left(12\right)$
Signatures:
sts

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