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11个4阶环例：

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cjsh
发表于 2011-6-30 15:03 | 显示全部楼层 |阅读模式
1 Rings with additive group Z4 Ring 4.U.1 Factor ring: Z/4Z={0,1,2,3} Unity Commutative One pair of zero divisors: (2)(2)=0 No nontrivial idempotents z:=4; R:=IntegerRing(z) ; R; exists(t){x : x in R | x^2 eq 0}; Characteristic(R); # R; FactoredModulus(R); Center(R); sub< R | 2 > ; Set(R); sub< R | 1 > ; Set(R); sub< R | 4 > ; Set(R); Modulus(R); Category(R); PrimeRing(R); AdditiveGroup(R); MultiplicativeGroup(R) ; UnitGroup(R) ; IsCommutative(R) ; IsUnitary(R); IsFinite(R); IsOrdered(R); IsField(R) ; IsEuclideanDomain(R); IsPID(R); IsUFD(R); IsDivisionRing(R); IsEuclideanRing(R) ; IsPrincipalIdealRing(R); IsDomain(R) ; Residue class ring of integers modulo 4 true 4 4 [ <2, 2> ] Residue class ring of integers modulo 4 Ideal of residue class ring of integers modulo 4 generated by 2 { 0, 1, 2, 3 } Residue class ring of integers modulo 4 { 0, 1, 2, 3 } Ideal of residue class ring of integers modulo 4 generated by 0 { 0, 1, 2, 3 } 4 RngIntRes Residue class ring of integers modulo 4 Abelian Group isomorphic to Z/4 Defined on 1 generator Relations: 4*$.1 = 0 Abelian Group isomorphic to Z/2 Defined on 1 generator Relations: 2*$.1 = 0 Abelian Group isomorphic to Z/2 Defined on 1 generator Relations: 2*$.1 = 0 true true true 4 false false false false false false true true false
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cjsh
 楼主| 发表于 2011-6-30 15:43 | 显示全部楼层

11个4阶环例：

2
Matrix ring: With coefficients from Z4,  0 0  
  0 0  
, a=   0 0  
  1 0  

,   0 0  
  2 0  
,   0 0  
  3 0  

Generated by a; a2=0 ("trivial multiplication")
Commutative
No unity
Every pair is a zero-divisor pair
No nontrivial idempotents
z:=4;
R1:=IntegerRing(z) ;
R1;
R := Matrix(R1, 4, 4, [0,0,0,0, 0,0,1,0, 0,0,2,0, 0,0,3,0]);
R ;
R*R;
R*R*R;
R ^ 4;
A ^ 5;
Parent(R);
Rank(R);
BaseRing(R);
CoefficientRing(R);
ElementToSequence(R);
Eltseq(R);
RowSequence(R) ;
Density(R);
Ncols(R);
NumberOfNonZeroEntries(R);
Submatrix(R, 4,4, 0,0);
Nullspace(R);
Kernel(R);
R ^ -1 ;
Transpose(R);
IsUnit(R);
IsSingular(R);
Trace(R);
TraceOfProduct(R ,R);
MinimalPolynomial(R);
CharacteristicPolynomial(R);
Adjoint(R);

Residue class ring of integers modulo 4
[0 0 0 0]
[0 0 1 0]
[0 0 2 0]
[0 0 3 0]
[0 0 0 0]
[0 0 2 0]
[0 0 0 0]
[0 0 2 0]
[0 0 0 0]
[0 0 0 0]
[0 0 0 0]
[0 0 0 0]
[0 0 0 0]
[0 0 0 0]
[0 0 0 0]
[0 0 0 0]
>> A ^ 5;
   ^
User error: Identifier ';A'; has not been declared or assigned
Full Matrix Algebra of degree 4 over IntegerRing(4)
1
Residue class ring of integers modulo 4
Residue class ring of integers modulo 4
[ 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 2, 0, 0, 0, 3, 0 ]
[ 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 2, 0, 0, 0, 3, 0 ]
[
    [ 0, 0, 0, 0 ],
    [ 0, 0, 1, 0 ],
    [ 0, 0, 2, 0 ],
    [ 0, 0, 3, 0 ]
]
0.187500000000000000000000000000
4
3
Matrix with 0 rows and 0 columns
RSpace of degree 4, dimension 3 over IntegerRing(4)
Echelonized basis:
(1 0 0 0)
(0 1 0 1)
(0 0 1 2)
RSpace of degree 4, dimension 3 over IntegerRing(4)
Echelonized basis:
(1 0 0 0)
(0 1 0 1)
(0 0 1 2)
>> R ^ -1 ;
     ^
Runtime error in ';^';: Argument 1 is not invertible
[0 0 0 0]
[0 0 0 0]
[0 1 2 3]
[0 0 0 0]
false
true
2
0
>> MinimalPolynomial(R);
                    ^
Runtime error in ';MinimalPolynomial';: Coefficient ring must be Z or an exact
field
&#36;.1^4 + 2*&#36;.1^3
[0 0 0 0]
[0 0 0 0]
[0 0 0 0]
[0 0 0 0]
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cjsh
 楼主| 发表于 2011-6-30 15:47 | 显示全部楼层

11个4阶环例：

3
As a subring of Z8: {0,2,4,6}
Matrix ring: With coefficients from Z4,  0 0  
  0 0  
, a=   1 1  
  1 1  

,   2 2  
  2 2  
,   3 3  
  3 3  

Generated by a; a2=2a
Commutative
No unity
Five pairs of zero divisors
No nontrivial idempotents

z:=8;
R1:=IntegerRing(z) ;
R1;
R := Matrix(R1, 4, 4, [0,0,0,0, 1,1,1,1, 2,2,2,2, 3,3,3,3]);
R ;
R*R;
R*R*R;
R ^ 4;
A ^ 5;
Parent(R);
Rank(R);
BaseRing(R);
CoefficientRing(R);
ElementToSequence(R);
Eltseq(R);
RowSequence(R) ;
Density(R);
Ncols(R);
NumberOfNonZeroEntries(R);
Submatrix(R, 4,4, 0,0);
Nullspace(R);
Kernel(R);
R ^ -1 ;
Transpose(R);
IsUnit(R);
IsSingular(R);
Trace(R);
TraceOfProduct(R ,R);
CharacteristicPolynomial(R);
Adjoint(R);

Residue class ring of integers modulo 8
[0 0 0 0]
[1 1 1 1]
[2 2 2 2]
[3 3 3 3]
[0 0 0 0]
[6 6 6 6]
[4 4 4 4]
[2 2 2 2]
[0 0 0 0]
[4 4 4 4]
[0 0 0 0]
[4 4 4 4]
[0 0 0 0]
[0 0 0 0]
[0 0 0 0]
[0 0 0 0]
>> A ^ 5;
   ^
User error: Identifier ';A'; has not been declared or assigned
Full Matrix Algebra of degree 4 over IntegerRing(8)
1
Residue class ring of integers modulo 8
Residue class ring of integers modulo 8
[ 0, 0, 0, 0, 1, 1, 1, 1, 2, 2, 2, 2, 3, 3, 3, 3 ]
[ 0, 0, 0, 0, 1, 1, 1, 1, 2, 2, 2, 2, 3, 3, 3, 3 ]
[
    [ 0, 0, 0, 0 ],
    [ 1, 1, 1, 1 ],
    [ 2, 2, 2, 2 ],
    [ 3, 3, 3, 3 ]
]
0.750000000000000000000000000000
4
12
Matrix with 0 rows and 0 columns
RSpace of degree 4, dimension 3 over IntegerRing(8)
Echelonized basis:
(1 0 0 0)
(0 1 0 5)
(0 0 1 2)
RSpace of degree 4, dimension 3 over IntegerRing(8)
Echelonized basis:
(1 0 0 0)
(0 1 0 5)
(0 0 1 2)
>> R ^ -1 ;
     ^
Runtime error in ';^';: Argument 1 is not invertible
[0 1 2 3]
[0 1 2 3]
[0 1 2 3]
[0 1 2 3]
false
true
6
4
&#36;.1^4 + 2*&#36;.1^3
[0 0 0 0]
[0 0 0 0]
[0 0 0 0]
[0 0 0 0]
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cjsh
 楼主| 发表于 2011-6-30 15:50 | 显示全部楼层

11个4阶环例：

4
Matrix presentation: With coefficients from Z4,  0 0  
  0 0  
,   2 0  
  0 0  
,   0 0  
  0 2  
,   2 0  
  0 2  

No unity
Trivial multiplication
No nontrivial idempotents
z:=4;
R1:=IntegerRing(z) ;
R1;
R := Matrix(R1, 4, 4, [0,0,0,0, 2,0,0,0, 0,0,0,2, 2,0,0,2]);
R ;
R*R;
R*R*R;
R ^ 4;
A ^ 5;
Parent(R);
Rank(R);
BaseRing(R);
CoefficientRing(R);
ElementToSequence(R);
Eltseq(R);
RowSequence(R) ;
Density(R);
Ncols(R);
NumberOfNonZeroEntries(R);
Submatrix(R, 4,4, 0,0);
Nullspace(R);
Kernel(R);
R ^ -1 ;
Transpose(R);
IsUnit(R);
IsSingular(R);
Trace(R);
TraceOfProduct(R ,R);
CharacteristicPolynomial(R);
Adjoint(R);
Residue class ring of integers modulo 4
[0 0 0 0]
[2 0 0 0]
[0 0 0 2]
[2 0 0 2]
[0 0 0 0]
[0 0 0 0]
[0 0 0 0]
[0 0 0 0]
[0 0 0 0]
[0 0 0 0]
[0 0 0 0]
[0 0 0 0]
[0 0 0 0]
[0 0 0 0]
[0 0 0 0]
[0 0 0 0]
>> A ^ 5;
   ^
User error: Identifier ';A'; has not been declared or assigned
Full Matrix Algebra of degree 4 over IntegerRing(4)
2
Residue class ring of integers modulo 4
Residue class ring of integers modulo 4
[ 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 2, 2, 0, 0, 2 ]
[ 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 2, 2, 0, 0, 2 ]
[
    [ 0, 0, 0, 0 ],
    [ 2, 0, 0, 0 ],
    [ 0, 0, 0, 2 ],
    [ 2, 0, 0, 2 ]
]
0.250000000000000000000000000000
4
4
Matrix with 0 rows and 0 columns
RSpace of degree 4, dimension 4 over IntegerRing(4)
Echelonized basis:
(1 0 0 0)
(0 1 1 1)
(0 0 2 0)
(0 0 0 2)
RSpace of degree 4, dimension 4 over IntegerRing(4)
Echelonized basis:
(1 0 0 0)
(0 1 1 1)
(0 0 2 0)
(0 0 0 2)
>> R ^ -1 ;
     ^
Runtime error in ';^';: Argument 1 is not invertible
[0 2 0 2]
[0 0 0 0]
[0 0 0 0]
[0 0 2 2]
false
true
2
0
&#36;.1^4 + 2*&#36;.1^3
[0 0 0 0]
[0 0 0 0]
[0 0 0 0]
[0 0 0 0]

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cjsh
 楼主| 发表于 2011-6-30 15:53 | 显示全部楼层

11个4阶环例：

[这个贴子最后由cjsh在 2011/07/01 11:58am 第 2 次编辑]

Non-commutative rings with additive group Z2+Z2

5

Matrix ring: With coefficients from Z2,  0 0  
  0 0  
, a=   1 0  
  0 0  

, b=   0 1  
  0 0  

,   1 1  
  0 0  

No multiplicative identity
Non-commutative (ab = b but ba = 0)
Two left-identities (a and a+b) but no right-identities.
Another representation as a matrix ring (coefficients in Z2):  0 0  
  0 0  
, a=   1 0  
  1 0  

, b=   1 1  
  1 1  

,   0 1  
  0 1  

A final representation:  0 0  
  0 0  
, a=   0 0  
  0 1  

, b=   0 0  
  1 0  

,   0 0  
  1 1  

z:=2;
R1:=IntegerRing(z) ;
R1;
R := Matrix(R1, 4, 4, [0,0,0,0, 1,0,0,0, 0,1,0,0, 1,1,0,0]);
R ;
R*R;
R*R*R;
R ^ 4;
A ^ 5;
Parent(R);
Rank(R);
BaseRing(R);
CoefficientRing(R);
ElementToSequence(R);
Eltseq(R);
RowSequence(R) ;
Density(R);
Ncols(R);
NumberOfNonZeroEntries(R);
Submatrix(R, 4,4, 0,0);
Nullspace(R);
Kernel(R);
R ^ -1 ;
Transpose(R);
IsUnit(R);
IsSingular(R);
Trace(R);
TraceOfProduct(R ,R);
CharacteristicPolynomial(R);
Adjoint(R);

Residue class ring of integers modulo 2
[0 0 0 0]
[1 0 0 0]
[0 1 0 0]
[1 1 0 0]
[0 0 0 0]
[0 0 0 0]
[1 0 0 0]
[1 0 0 0]
[0 0 0 0]
[0 0 0 0]
[0 0 0 0]
[0 0 0 0]
[0 0 0 0]
[0 0 0 0]
[0 0 0 0]
[0 0 0 0]
>> A ^ 5;
   ^
User error: Identifier ';A'; has not been declared or assigned
Full Matrix Algebra of degree 4 over IntegerRing(2)
2
Residue class ring of integers modulo 2
Residue class ring of integers modulo 2
[ 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 1, 1, 0, 0 ]
[ 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 1, 1, 0, 0 ]
[
    [ 0, 0, 0, 0 ],
    [ 1, 0, 0, 0 ],
    [ 0, 1, 0, 0 ],
    [ 1, 1, 0, 0 ]
]
0.250000000000000000000000000000
4
4
Matrix with 0 rows and 0 columns
Vector space of degree 4, dimension 2 over IntegerRing(2)
Echelonized basis:
(1 0 0 0)
(0 1 1 1)
Vector space of degree 4, dimension 2 over IntegerRing(2)
Echelonized basis:
(1 0 0 0)
(0 1 1 1)
>> R ^ -1 ;
     ^
Runtime error in ';^';: Argument 1 is not invertible
[0 1 0 1]
[0 0 1 1]
[0 0 0 0]
[0 0 0 0]
false
true
0
0
&#36;.1^4
[0 0 0 0]
[0 0 0 0]
[0 0 0 0]
[0 0 0 0]

z:=2;
R1:=IntegerRing(z) ;
R1;
R := Matrix(R1, 4, 4, [0,0,0,0, 1,0,1,0, 1,1,1,1, 0,1,0,1]);
R ;
R*R;
R*R*R;
R ^ 4;
A ^ 5;
Parent(R);
Rank(R);
BaseRing(R);
CoefficientRing(R);
ElementToSequence(R);
Eltseq(R);
RowSequence(R) ;
Density(R);
Ncols(R);
NumberOfNonZeroEntries(R);
Submatrix(R, 4,4, 0,0);
Nullspace(R);
Kernel(R);
R ^ -1 ;
Transpose(R);
IsUnit(R);
IsSingular(R);
Trace(R);
TraceOfProduct(R ,R);
CharacteristicPolynomial(R);
Adjoint(R);

Residue class ring of integers modulo 2
[0 0 0 0]
[1 0 1 0]
[1 1 1 1]
[0 1 0 1]
[0 0 0 0]
[1 1 1 1]
[0 0 0 0]
[1 1 1 1]
[0 0 0 0]
[0 0 0 0]
[0 0 0 0]
[0 0 0 0]
[0 0 0 0]
[0 0 0 0]
[0 0 0 0]
[0 0 0 0]
>> A ^ 5;
   ^
User error: Identifier ';A'; has not been declared or assigned
Full Matrix Algebra of degree 4 over IntegerRing(2)
2
Residue class ring of integers modulo 2
Residue class ring of integers modulo 2
[ 0, 0, 0, 0, 1, 0, 1, 0, 1, 1, 1, 1, 0, 1, 0, 1 ]
[ 0, 0, 0, 0, 1, 0, 1, 0, 1, 1, 1, 1, 0, 1, 0, 1 ]
[
    [ 0, 0, 0, 0 ],
    [ 1, 0, 1, 0 ],
    [ 1, 1, 1, 1 ],
    [ 0, 1, 0, 1 ]
]
0.500000000000000000000000000000
4
8
Matrix with 0 rows and 0 columns
Vector space of degree 4, dimension 2 over IntegerRing(2)
Echelonized basis:
(1 0 0 0)
(0 1 1 1)
Vector space of degree 4, dimension 2 over IntegerRing(2)
Echelonized basis:
(1 0 0 0)
(0 1 1 1)
>> R ^ -1 ;
     ^
Runtime error in ';^';: Argument 1 is not invertible
[0 1 1 0]
[0 0 1 1]
[0 1 1 0]
[0 0 1 1]
false
true
0
0
&#36;.1^4
[0 0 0 0]
[0 0 0 0]
[0 0 0 0]
[0 0 0 0]





z:=2;
R1:=IntegerRing(z) ;
R1;
R := Matrix(R1, 4, 4, [0,0,0,0, 0,0,0,1, 0,0,1,0, 0,0,1,1]);
R ;
R*R;
R*R*R;
R ^ 4;
A ^ 5;
Parent(R);
Rank(R);
BaseRing(R);
CoefficientRing(R);
ElementToSequence(R);
Eltseq(R);
RowSequence(R) ;
Density(R);
Ncols(R);
NumberOfNonZeroEntries(R);
Submatrix(R, 4,4, 0,0);
Nullspace(R);
Kernel(R);
R ^ -1 ;
Transpose(R);
IsUnit(R);
IsSingular(R);
Trace(R);
TraceOfProduct(R ,R);
CharacteristicPolynomial(R);
Adjoint(R);
Residue class ring of integers modulo 2
[0 0 0 0]
[0 0 0 1]
[0 0 1 0]
[0 0 1 1]
[0 0 0 0]
[0 0 1 1]
[0 0 1 0]
[0 0 0 1]
[0 0 0 0]
[0 0 0 1]
[0 0 1 0]
[0 0 1 1]
[0 0 0 0]
[0 0 1 1]
[0 0 1 0]
[0 0 0 1]
>> A ^ 5;
   ^
User error: Identifier ';A'; has not been declared or assigned
Full Matrix Algebra of degree 4 over IntegerRing(2)
2
Residue class ring of integers modulo 2
Residue class ring of integers modulo 2
[ 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 1 ]
[ 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 1 ]
[
    [ 0, 0, 0, 0 ],
    [ 0, 0, 0, 1 ],
    [ 0, 0, 1, 0 ],
    [ 0, 0, 1, 1 ]
]
0.250000000000000000000000000000
4
4
Matrix with 0 rows and 0 columns
Vector space of degree 4, dimension 2 over IntegerRing(2)
Echelonized basis:
(1 0 0 0)
(0 1 1 1)
Vector space of degree 4, dimension 2 over IntegerRing(2)
Echelonized basis:
(1 0 0 0)
(0 1 1 1)
>> R ^ -1 ;
     ^
Runtime error in ';^';: Argument 1 is not invertible
[0 0 0 0]
[0 0 0 0]
[0 0 1 1]
[0 1 0 1]
false
true
0
0
&#36;.1^4 + &#36;.1^2
[0 0 0 0]
[0 0 0 0]
[0 0 0 0]
[0 0 0 0]

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cjsh
 楼主| 发表于 2011-6-30 15:53 | 显示全部楼层

11个4阶环例：

还有6种，等明天。。
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cjsh
 楼主| 发表于 2011-7-1 11:55 | 显示全部楼层

11个4阶环例：

[这个贴子最后由cjsh在 2011/07/01 00:02pm 第 2 次编辑]

Non-commutative rings with additive group Z2+Z2

6

Ring 22.NC.2
Matrix ring: With coefficients from Z2,  0 0  
  0 0  
, a=   1 0  
  0 0  

, b=   0 0  
  1 0  

,   1 0  
  1 0  

No multiplicative identity
Non-commutative (ab = 0 but ba = b)
Two right-identities (a and a+b) but no left-identities.
Another representation as a matrix ring (coefficients in Z2):  0 0  
  0 0  
, a=   0 0  
  1 1  

, b=   1 1  
  1 1  

,   1 1  
  0 0  

A final representation:  0 0  
  0 0  
, a=   0 0  
  0 1  

, b=   0 1  
  0 0  

,   0 1  
  0 1  


z:=2;
R1:=IntegerRing(z) ;
R1;
R := Matrix(R1, 4, 4, [0,0,0,0, 1,0,0,0, 0,0,1,0, 1,0,1,0]);
R ;
R2:=R*R;
R2;
R3:=R2*R;
R3;
R33:=R*R2;
R33;
R4:=R3*R;
R4;
R44:=R*R3;
R44;
R ^ 4;
A ^ 5;
Parent(R);
Rank(R);
BaseRing(R);
CoefficientRing(R);
ElementToSequence(R);
Eltseq(R);
RowSequence(R) ;
Density(R);
Ncols(R);
NumberOfNonZeroEntries(R);
Submatrix(R, 4,4, 0,0);
Nullspace(R);
Kernel(R);
R ^ -1 ;
Transpose(R);
IsUnit(R);
IsSingular(R);
Trace(R);
TraceOfProduct(R ,R);
CharacteristicPolynomial(R);
Adjoint(R);
Residue class ring of integers modulo 2
[0 0 0 0]
[1 0 0 0]
[0 0 1 0]
[1 0 1 0]
[0 0 0 0]
[0 0 0 0]
[0 0 1 0]
[0 0 1 0]
[0 0 0 0]
[0 0 0 0]
[0 0 1 0]
[0 0 1 0]
[0 0 0 0]
[0 0 0 0]
[0 0 1 0]
[0 0 1 0]
[0 0 0 0]
[0 0 0 0]
[0 0 1 0]
[0 0 1 0]
[0 0 0 0]
[0 0 0 0]
[0 0 1 0]
[0 0 1 0]
[0 0 0 0]
[0 0 0 0]
[0 0 1 0]
[0 0 1 0]
>> A ^ 5;
   ^
User error: Identifier ';A'; has not been declared or assigned
Full Matrix Algebra of degree 4 over IntegerRing(2)
2
Residue class ring of integers modulo 2
Residue class ring of integers modulo 2
[ 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 1, 0, 1, 0 ]
[ 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 1, 0, 1, 0 ]
[
    [ 0, 0, 0, 0 ],
    [ 1, 0, 0, 0 ],
    [ 0, 0, 1, 0 ],
    [ 1, 0, 1, 0 ]
]
0.250000000000000000000000000000
4
4
Matrix with 0 rows and 0 columns
Vector space of degree 4, dimension 2 over IntegerRing(2)
Echelonized basis:
(1 0 0 0)
(0 1 1 1)
Vector space of degree 4, dimension 2 over IntegerRing(2)
Echelonized basis:
(1 0 0 0)
(0 1 1 1)
>> R ^ -1 ;
     ^
Runtime error in ';^';: Argument 1 is not invertible
[0 1 0 1]
[0 0 0 0]
[0 0 1 1]
[0 0 0 0]
false
true
1
1
&#36;.1^4 + &#36;.1^3
[0 0 0 0]
[0 0 0 0]
[0 0 0 0]
[0 0 0 0]

z:=2;
R1:=IntegerRing(z) ;
R1;
R := Matrix(R1, 4, 4, [0,0,0,0, 0,0,1,1, 1,1,1,1, 1,1,0,0]);
R ;
R2:=R*R;
R2;
R3:=R2*R;
R3;
R33:=R*R2;
R33;
R4:=R3*R;
R4;
R44:=R*R3;
R44;
R ^ 4;
A ^ 5;
Parent(R);
Rank(R);
BaseRing(R);
CoefficientRing(R);
ElementToSequence(R);
Eltseq(R);
RowSequence(R) ;
Density(R);
Ncols(R);
NumberOfNonZeroEntries(R);
Submatrix(R, 4,4, 0,0);
Nullspace(R);
Kernel(R);
R ^ -1 ;
Transpose(R);
IsUnit(R);
IsSingular(R);
Trace(R);
TraceOfProduct(R ,R);
CharacteristicPolynomial(R);
Adjoint(R);

Residue class ring of integers modulo 2
[0 0 0 0]
[0 0 1 1]
[1 1 1 1]
[1 1 0 0]
[0 0 0 0]
[0 0 1 1]
[0 0 0 0]
[0 0 1 1]
[0 0 0 0]
[0 0 1 1]
[0 0 0 0]
[0 0 1 1]
[0 0 0 0]
[0 0 1 1]
[0 0 0 0]
[0 0 1 1]
[0 0 0 0]
[0 0 1 1]
[0 0 0 0]
[0 0 1 1]
[0 0 0 0]
[0 0 1 1]
[0 0 0 0]
[0 0 1 1]
[0 0 0 0]
[0 0 1 1]
[0 0 0 0]
[0 0 1 1]
>> A ^ 5;
   ^
User error: Identifier ';A'; has not been declared or assigned
Full Matrix Algebra of degree 4 over IntegerRing(2)
2
Residue class ring of integers modulo 2
Residue class ring of integers modulo 2
[ 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 0, 0 ]
[ 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 0, 0 ]
[
    [ 0, 0, 0, 0 ],
    [ 0, 0, 1, 1 ],
    [ 1, 1, 1, 1 ],
    [ 1, 1, 0, 0 ]
]
0.500000000000000000000000000000
4
8
Matrix with 0 rows and 0 columns
Vector space of degree 4, dimension 2 over IntegerRing(2)
Echelonized basis:
(1 0 0 0)
(0 1 1 1)
Vector space of degree 4, dimension 2 over IntegerRing(2)
Echelonized basis:
(1 0 0 0)
(0 1 1 1)
>> R ^ -1 ;
     ^
Runtime error in ';^';: Argument 1 is not invertible
[0 0 1 1]
[0 0 1 1]
[0 1 1 0]
[0 1 1 0]
false
true
1
1
&#36;.1^4 + &#36;.1^3
[0 0 0 0]
[0 0 0 0]
[0 0 0 0]
[0 0 0 0]

z:=2;
R1:=IntegerRing(z) ;
R1;
R := Matrix(R1, 4, 4, [0,0,0,0, 0,0,0,1, 0,1,0,0, 0,1,0,1]);
R ;
R2:=R*R;
R2;
R3:=R2*R;
R3;
R33:=R*R2;
R33;
R4:=R3*R;
R4;
R44:=R*R3;
R44;
R ^ 4;
A ^ 5;
Parent(R);
Rank(R);
BaseRing(R);
CoefficientRing(R);
ElementToSequence(R);
Eltseq(R);
RowSequence(R) ;
Density(R);
Ncols(R);
NumberOfNonZeroEntries(R);
Submatrix(R, 4,4, 0,0);
Nullspace(R);
Kernel(R);
R ^ -1 ;
Transpose(R);
IsUnit(R);
IsSingular(R);
Trace(R);
TraceOfProduct(R ,R);
CharacteristicPolynomial(R);
Adjoint(R);

Residue class ring of integers modulo 2
[0 0 0 0]
[0 0 0 1]
[0 1 0 0]
[0 1 0 1]
[0 0 0 0]
[0 1 0 1]
[0 0 0 1]
[0 1 0 0]
[0 0 0 0]
[0 1 0 0]
[0 1 0 1]
[0 0 0 1]
[0 0 0 0]
[0 1 0 0]
[0 1 0 1]
[0 0 0 1]
[0 0 0 0]
[0 0 0 1]
[0 1 0 0]
[0 1 0 1]
[0 0 0 0]
[0 0 0 1]
[0 1 0 0]
[0 1 0 1]
[0 0 0 0]
[0 0 0 1]
[0 1 0 0]
[0 1 0 1]
>> A ^ 5;
   ^
User error: Identifier ';A'; has not been declared or assigned
Full Matrix Algebra of degree 4 over IntegerRing(2)
2
Residue class ring of integers modulo 2
Residue class ring of integers modulo 2
[ 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 1, 0, 1 ]
[ 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 1, 0, 1 ]
[
    [ 0, 0, 0, 0 ],
    [ 0, 0, 0, 1 ],
    [ 0, 1, 0, 0 ],
    [ 0, 1, 0, 1 ]
]
0.250000000000000000000000000000
4
4
Matrix with 0 rows and 0 columns
Vector space of degree 4, dimension 2 over IntegerRing(2)
Echelonized basis:
(1 0 0 0)
(0 1 1 1)
Vector space of degree 4, dimension 2 over IntegerRing(2)
Echelonized basis:
(1 0 0 0)
(0 1 1 1)
>> R ^ -1 ;
     ^
Runtime error in ';^';: Argument 1 is not invertible
[0 0 0 0]
[0 0 1 1]
[0 0 0 0]
[0 1 0 1]
false
true
1
1
&#36;.1^4 + &#36;.1^3 + &#36;.1^2
[0 0 0 0]
[0 0 0 0]
[0 0 0 0]
[0 0 0 0]
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cjsh
 楼主| 发表于 2011-7-1 12:03 | 显示全部楼层

11个4阶环例：

Commutative rings with additive group Z2+Z2, with unit Ring 22.U.1 Ring direct sum: Z2+Z2 Factor ring: Z2[x]/ Matrix presentation: With coefficients from Z2, 0 0 0 0 , 1 0 0 1 , 1 0 0 0 , 0 0 0 1 Unity Two pairs of zero divisors Two nontrivial idempotents Ring 22.U.2 Factor Ring: Z2[x]/={0,1,x,1+x} Matrix Ring: With coefficients from Z2, 0 0 0 0 , 1 0 0 1 , 0 1 1 0 , 1 1 1 1 Unity One pair of zero divisors: (x+1)(x+1)=0 No nontrivial idempotents Ring 22.U.3 Factor ring: Z2[x]/={0,1,x,1+x} Matrix ring: With coefficients from Z2, 0 0 0 0 , 1= 1 0 0 1 , x= 1 1 1 0 , 0 1 1 1 Unity No zero divisors (it';s a field) No nontrivial idempotents
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