OpenMath Content Dictionary: ringname1

Canonical URL:

http://www.openmath.org/cd/ringname1.ocd

CD File:

ringname1.ocd

CD as XML Encoded OpenMath:

ringname1.omcd

Defines:

quaternions, Z, Zm

Date:

2004-03-08

Version:

1

Review Date:

2005-04-01

Status:

experimental

Uses CD:

alg1, arith1, fns1, fns2, group1, group2, logic1, monoid1, monoid2, nums1, quant1, relation1, set1, setname1, ring1

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):

xml prefix mathml

<OMOBJ xmlns="http://www.openmath.org/OpenMath" version="2.0">
  <OMA><OMS name="eq" cd="relation1"/>
       <OMA><OMS name="identity" cd="ring1"/>
            <OMS name="Z" cd="ringname1"/>
       </OMA>
       <OMI>1</OMI>
  </OMA>
</OMOBJ>

eq (identity (Z) , 1)

$$\mathrm{identity}\left(Z\right)=1$$

Commented Mathematical property (CMP):

The carrier set of this ring is the set of integers.

Formal Mathematical property (FMP):

xml prefix mathml

<OMOBJ xmlns="http://www.openmath.org/OpenMath" version="2.0">
  <OMA><OMS name="eq" cd="relation1"/>
       <OMA><OMS name="carrier" cd="ring1"/>
            <OMS name="Z" cd="ringname1"/>
       </OMA>
       <OMS name="Z" cd="setname1"/>
  </OMA>
</OMOBJ>

eq (carrier (Z) , Z)

$$\mathrm{carrier}\left(Z\right)=\mathbb{Z}$$

Signatures:

sts

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):

xml prefix mathml

<OMOBJ xmlns="http://www.openmath.org/OpenMath" version="2.0">
  <OMA><OMS cd="ring2" name="isomorphic"/>
       <OMA><OMS cd="ringname1" name="quaternions"/>
            <OMV name="R"/>
       </OMA>
       <OMA><OMS cd="ring3" name="quotient_ring"/>
            <OMA id="fr"><OMS cd="ring3" name="free_ring"/>
                 <OMS cd="fieldname1" name="Q"/>
                 <OMV name="i"/> <OMV name="j"/>  <OMV name="k"/>
            </OMA>
            <OMA><OMS cd="ring3" name="ideal"/>
                 <OMR href="#fr"/>
                 <OMA><OMS cd="list1" name="list"/>
                      <OMA><OMS cd="arith1" name="minus"/>
                           <OMA><OMS cd="arith1" name="times"/>
                                <OMV name="i"/> <OMV name="j"/>  
                           </OMA>
                           <OMV name="k"/>
                      </OMA>
                      <OMA><OMS cd="arith1" name="plus"/>
                           <OMA><OMS cd="arith1" name="times"/>
                                <OMV name="j"/> <OMV name="i"/>  
                           </OMA>
                           <OMV name="k"/>
                      </OMA>
                      <OMA><OMS cd="arith1" name="plus"/>
                           <OMA><OMS cd="arith1" name="power"/>
                                <OMV name="i"/>  
                                <OMI>2</OMI>
                           </OMA>
                                <OMI>1</OMI>
                      </OMA>
                      <OMA><OMS cd="arith1" name="plus"/>
                           <OMA><OMS cd="arith1" name="power"/>
                                <OMV name="j"/>  
                                <OMI>2</OMI>
                           </OMA>
                                <OMI>1</OMI>
                      </OMA>
                      <OMA><OMS cd="arith1" name="plus"/>
                           <OMA><OMS cd="arith1" name="power"/>
                                <OMV name="k"/>  
                                <OMI>2</OMI>
                           </OMA>
                                <OMI>1</OMI>
                      </OMA>
                </OMA>
           </OMA>
      </OMA>
 </OMA>
</OMOBJ>

isomorphic (quaternions ( R) , quotient_ring (free_ring (Q, i, j, k) , ideal (, list (minus (times ( i, j) , k) , plus (times ( j, i) , k) , plus (power ( i, 2) , 1) , plus (power ( j, 2) , 1) , plus (power ( k, 2) , 1) ) ) ) )

$$\mathrm{isomorphic}(\mathrm{quaternions}\left(R\right),\mathrm{quotient\_ring}(\mathrm{free\_ring}(Q,i,j,k),\mathrm{ideal}(,\begin{array}{c}ij-k\hfill \\ ji+k\hfill \\ i^2+1\hfill \\ j^2+1\hfill \\ k^2+1\hfill \end{array})))$$

Signatures:

sts

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:

xml prefix mathml

<OMOBJ xmlns="http://www.openmath.org/OpenMath" version="2.0" cdbase="http://www.openmath.org/cd">
  <OMA>
    <OMS name="Zm" cd="ringname1"/>
    <OMI> 12 </OMI>
  </OMA>
</OMOBJ>

Zm ( 12 )

$$\mathrm{Zm}\left(12\right)$$

Signatures:

sts