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

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:

xml prefix mathml

<OMOBJ xmlns="http://www.openmath.org/OpenMath" version="2.0">
  <OMA><OMS cd="ring3" name="ideal"/>
       <OMA><OMS cd="ring3" name="free_ring"/>
            <OMA><OMS cd="list1" name="list"/>
                 <OMV name="a"/>  <OMV name="b"/>
            </OMA>
       </OMA>
       <OMA><OMS cd="list1" name="list"/>
            <OMA><OMS cd="arith1" name="minus"/>
                 <OMA><OMS cd="arith1" name="times"/>
                      <OMV name="a"/>  <OMV name="b"/>
                 </OMA>
                 <OMA><OMS cd="arith1" name="times"/>
                      <OMV name="b"/>  <OMV name="a"/>
                 </OMA>
            </OMA>
       </OMA>
  </OMA>
</OMOBJ>

ideal (free_ring (list ( a, b) ) , list (minus (times ( a, b) , times ( b, a) ) ) )

$$\mathrm{ideal}(\mathrm{free\_ring}\left((a,b)\right),\begin{array}{c}ab-ba\hfill \end{array})$$

Signatures:

sts

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):

xml prefix mathml

<OMOBJ xmlns="http://www.openmath.org/OpenMath" version="2.0">
  <OMA><OMS cd="logic1" name="implies"/>
       <OMA><OMS cd="ring2" name="is_homomorphism"/>
            <OMV name="R"/>  <OMV name="S"/>   <OMV name="f"/> 
       </OMA>
       <OMA><OMS cd="ring3" name="is_ideal"/>
            <OMV name="R"/>
            <OMA><OMS cd="ring3" name="kernel"/>
                 <OMV name="f"/>
            </OMA>
       </OMA>
  </OMA>
</OMOBJ>

implies (is_homomorphism ( R, S, f) , is_ideal ( R, kernel ( f) ) )

$$\mathrm{is\_homomorphism}(R,S,f)\Rightarrow \mathrm{is\_ideal}(R,\mathrm{kernel}\left(f\right))$$

Signatures:

sts

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:

xml prefix mathml

<OMOBJ xmlns="http://www.openmath.org/OpenMath" version="2.0">
  <OMA><OMS cd="ring3" name="principal_ideal"/>
       <OMA><OMS cd="ring3" name="free_ring"/>
            <OMS cd="fieldname1" name="Q"/>
            <OMA><OMS cd="list1" name="list"/>
                 <OMV name="a"/>  <OMV name="b"/>
            </OMA>
       </OMA>
       <OMA><OMS cd="arith1" name="minus"/>
            <OMA><OMS cd="arith1" name="times"/>
                 <OMV name="a"/>  <OMV name="b"/>
            </OMA>
            <OMA><OMS cd="arith1" name="times"/>
                 <OMV name="b"/>  <OMV name="a"/>
            </OMA>
       </OMA>
  </OMA>
</OMOBJ>

principal_ideal (free_ring (Q, list ( a, b) ) , minus (times ( a, b) , times ( b, a) ) )

$$\mathrm{principal\_ideal}(\mathrm{free\_ring}(Q,(a,b)),ab-ba)$$

Signatures:

sts

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:

xml prefix mathml

<OMOBJ xmlns="http://www.openmath.org/OpenMath" version="2.0">
  <OMA><OMS cd="ring3" name="free_ring"/>
       <OMV name="R"/>
       <OMA><OMS cd="list1" name="list"/>
            <OMV name="a"/>  <OMV name="b"/>
       </OMA>
  </OMA>
</OMOBJ>

free_ring ( R, list ( a, b) )

$$\mathrm{free\_ring}(R,(a,b))$$

Signatures:

sts

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:

xml prefix mathml

<OMOBJ xmlns="http://www.openmath.org/OpenMath" version="2.0">
  <OMA><OMS cd="ring3" name="poly_ring"/>
       <OMV name="R"/>
       <OMV name="X"/> 
  </OMA>
</OMOBJ>

poly_ring ( R, X)

$$\mathrm{poly\_ring}(R,X)$$

Signatures:

sts

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:

xml prefix mathml

<OMOBJ xmlns="http://www.openmath.org/OpenMath" version="2.0">
  <OMA><OMS cd="ring3" name="m_poly_ring"/>
       <OMV name="R"/>
       <OMA><OMS cd="list1" name="list"/>
            <OMV name="a"/>  <OMV name="b"/>
       </OMA>
  </OMA>
</OMOBJ>

m_poly_ring ( R, list ( a, b) )

$$\mathrm{m\_poly\_ring}(R,(a,b))$$

Signatures:

sts

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):

xml prefix mathml

<OMOBJ xmlns="http://www.openmath.org/OpenMath" version="2.0">
  <OMA><OMS cd="ring2" name="isomorphic"/>
       <OMA><OMS cd="ring3" name="matrix_ring"/>
            <OMI>1</OMI>  <OMV name="R"/>
       </OMA>
       <OMV name="R"/>
  </OMA>
</OMOBJ>

isomorphic (matrix_ring (1, R) , R)

$$\mathrm{isomorphic}(\mathrm{matrix\_ring}(1,R),R)$$

Signatures:

sts

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

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:

xml prefix mathml

<OMOBJ xmlns="http://www.openmath.org/OpenMath" version="2.0">
    <OMA><OMS cd="relation1" name="eq"/>
        <OMA><OMS cd="ring3" name="direct_product"/>
            <OMA><OMS cd="ring1" name="ring"/>
                <OMS cd="setname1" name="Z"/>
                <OMS cd="arith1" name="plus"/>
                <OMS cd="arith1" name="unary_minus"/>
                <OMI>0</OMI>
            </OMA>
            <OMA><OMS cd="ring1" name="ring"/>
                <OMS cd="setname1" name="Z"/>
                <OMS cd="arith1" name="plus"/>
                <OMS cd="arith1" name="unary_minus"/>
                <OMI>0</OMI>
            </OMA>
        </OMA>
        <OMA><OMS cd="ring3" name="direct_power"/>
            <OMA><OMS cd="ring1" name="ring"/>
                <OMS cd="setname1" name="Z"/>
                <OMS cd="arith1" name="plus"/>
                <OMS cd="arith1" name="unary_minus"/>
                <OMI>0</OMI>
            </OMA>
            <OMI>2</OMI>
        </OMA>
   </OMA>
  </OMOBJ>

eq (direct_product (ring (Z, plus, unary_minus, 0) , ring (Z, plus, unary_minus, 0) ) , direct_power (ring (Z, plus, unary_minus, 0) , 2) )

$$\mathrm{direct\_product}(\mathrm{ring}(\mathbb{Z},+,-,0),\mathrm{ring}(\mathbb{Z},+,-,0))=\mathrm{direct\_power}(\mathrm{ring}(\mathbb{Z},+,-,0),2)$$

Signatures:

sts

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.

xml prefix mathml

<OMOBJ xmlns="http://www.openmath.org/OpenMath" version="2.0">
          <OMA><OMS cd="ring3" name="quotient_ring"/><!-- (Z/2Z) -->
               <OMA><OMS cd="ring1" name="ring"/><!-- Z -->
                  <OMS cd="setname1" name="Z"/>
                  <OMS cd="arith1" name="plus"/>
                  <OMI>0</OMI>
                  <OMS cd="arith1" name="minus"/>
                  <OMS cd="arith1" name="times"/>
                  <OMI>1</OMI>
               </OMA>
               <OMA><OMS cd="ring3" name="ideal"/><!-- 2Z -->
                  <OMA><OMS cd="ring1" name="ring"/><!-- Z -->
                     <OMS cd="setname1" name="Z"/>
                     <OMS cd="arith1" name="plus"/>
                     <OMI>0</OMI>
                     <OMS cd="arith1" name="minus"/>
                     <OMS cd="arith1" name="times"/>
                     <OMI>1</OMI>
                  </OMA>
                  <OMA><OMS cd="list1" name="list"/>
                     <OMI>2</OMI>
                  </OMA>
               </OMA>
          </OMA>
</OMOBJ>

quotient_ring (ring (Z, plus, 0, minus, times, 1) , ideal (ring (Z, plus, 0, minus, times, 1) , list (2) ) )

$$\mathrm{quotient\_ring}(\mathrm{ring}(\mathbb{Z},+,0,-,\times ,1),\mathrm{ideal}(\mathrm{ring}(\mathbb{Z},+,0,-,\times ,1),\left(2\right)))$$

Example:

The ring (Z/2Z)[x]/(x^2+x+1)

xml prefix mathml

<OMOBJ xmlns="http://www.openmath.org/OpenMath" version="2.0">
   <OMA><OMS cd="ring3" name="quotient_ring"/><!-- (Z/2Z)[x]/(x^2+x+1) -->
      <OMA><OMS cd="ring3" name="poly_ring"/><!-- (Z/2Z)[x] -->
         <OMA><OMS cd="setname2" name="Zm"/><!-- (Z/2Z) -->
               <OMI>2</OMI>
         </OMA>
         <OMV name="x"/><!-- [x] -->
      </OMA>
      <OMA><OMS cd="ring3" name="ideal"/><!-- (x^2+x+1) -->
         <OMA><OMS cd="ring3" name="poly_ring"/><!-- (Z/2Z)[x] -->
              <OMA><OMS cd="setname2" name="Zm"/><!-- (Z/2Z) -->
                   <OMI>2</OMI>
              </OMA>
              <OMV name="x"/>
         </OMA>
         <OMA><OMS cd="list1" name="list"/>
            <OMA><OMS cd="arith1" name="plus"/>
               <OMA><OMS cd="arith1" name="power"/>
                  <OMV name="x"/>
                  <OMI>2</OMI>
               </OMA>
               <OMV name="x"/>
               <OMI>1</OMI>
            </OMA>
         </OMA>
      </OMA>
   </OMA>
</OMOBJ>

quotient_ring (poly_ring (Zm (2) , x) , ideal (poly_ring (Zm (2) , x) , list (plus (power ( x, 2) , x, 1) ) ) )

$$\mathrm{quotient\_ring}(\mathrm{poly\_ring}({\mathbb{Z}}_2,x),\mathrm{ideal}(\mathrm{poly\_ring}({\mathbb{Z}}_2,x),\begin{array}{c}{x}^2+x+1\hfill \end{array}))$$

Using the xref mechanism it can also be represented as

xml prefix mathml

<OMOBJ xmlns="http://www.openmath.org/OpenMath" version="2.0">
   <OMA><OMS cd="ring3" name="quotient_ring"/><!-- (Z/2Z)[x]/(x^2+x+1) -->
        <OMA id="domain"><OMS cd="ring3" name="poly_ring"/><!-- (Z/2Z)[x] -->
             <OMA><OMS cd="setname2" name="Zm"/><!-- (Z/2Z) -->
                  <OMI>2</OMI>
             </OMA>
             <OMV name="x"/><!-- [x] -->
        </OMA>
        <OMA><OMS cd="ring3" name="principal_ideal"/><!-- (x^2+x+1) -->
             <OMR href="#domain"/>
             <OMA><OMS cd="arith1" name="plus"/>
                  <OMA><OMS cd="arith1" name="power"/>
                       <OMV name="x"/> <OMI>2</OMI>
                  </OMA>
                  <OMV name="x"/>
                  <OMI>1</OMI>
             </OMA>
        </OMA>
   </OMA>
</OMOBJ>

quotient_ring (poly_ring (Zm (2) , x) , principal_ideal (, plus (power ( x, 2) , x, 1) ) )

$$\mathrm{quotient\_ring}(\mathrm{poly\_ring}({\mathbb{Z}}_2,x),\mathrm{principal\_ideal}(\mathrm{poly\_ring}({\mathbb{Z}}_2,x),x^2+x+1))$$

Signatures:

sts

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):

xml prefix mathml

<OMOBJ xmlns="http://www.openmath.org/OpenMath" version="2.0">
  <OMA><OMS cd="relation1" name="eq"/>
       <OMA><OMS cd="ring3" name="invertibles"/>
            <OMV name="R"/>
       </OMA>
       <OMA><OMS cd="group3" name="invertibles"/>
            <OMA><OMS cd="ring1" name="multiplicative_monoid"/>
                 <OMV name="R"/>
            </OMA>
       </OMA>
  </OMA>
</OMOBJ>

eq (invertibles ( R) , invertibles (multiplicative_monoid ( R) ) )

$$\mathrm{invertibles}\left(R\right)=\mathrm{invertibles}\left(\mathrm{multiplicative\_monoid}\left(R\right)\right)$$

Signatures:

sts

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):

xml prefix mathml

<OMOBJ xmlns="http://www.openmath.org/OpenMath" version="2.0">
  <OMA><OMS cd="relation1" name="eq"/>
       <OMA><OMS cd="ring3" name="invertibles"/>
            <OMV name="R"/>
       </OMA>
       <OMA><OMS cd="group1" name="carrier"/>
            <OMA><OMS cd="ring3" name="multiplicative_group"/>
                 <OMV name="R"/>
            </OMA>
       </OMA>
  </OMA>
</OMOBJ>

eq (invertibles ( R) , carrier (multiplicative_group ( R) ) )

$$\mathrm{invertibles}\left(R\right)=\mathrm{carrier}\left(\mathrm{multiplicative\_group}\left(R\right)\right)$$

Signatures:

sts

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):

xml prefix mathml

<OMOBJ xmlns="http://www.openmath.org/OpenMath" version="2.0">
  <OMA><OMS cd="relation1" name="eq"/>
       <OMS cd="ring3" name="integers"/>
       <OMA><OMS cd="ring1" name="ring"/>
            <OMA><OMS cd="setname1" name="Z"/>
                 <OMS cd="arith1" name="plus"/>
                 <OMI>0</OMI>
                 <OMS cd="arith1" name="minus"/>
                 <OMS cd="arith1" name="times"/>
                 <OMI>1</OMI>
            </OMA>
       </OMA>

  </OMA>
</OMOBJ>

eq (integers, ring (Z (plus, 0, minus, times, 1) ) )

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

Signatures:

sts