|

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
$.1^4 + $.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
$.1^4 + $.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
$.1^4 + $.1^3 + $.1^2
[0 0 0 0]
[0 0 0 0]
[0 0 0 0]
[0 0 0 0] |
|