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[这个贴子最后由cjsh在 2011/07/21 10:10am 第 1 次编辑]
反环和环的其它五项定义条件一样,除了乘法为r*s=sr
RING反同态为φ(rs)=φ(s)φ(r)的有单位元1的ABEL环
环到该环全部反同态集合的映射为左模
Z:=8;
R:=IntegerRing(Z) ;
R;
I3:=ideal< R | 3 > ;
I3;
I10:=ideal< R | 10 > ;
I10;
RR:=ResidueClassRing(Z);
RR;
RC:=Characteristic(R) ;
RC;
# R;
Modulus(R);
FactoredModulus(R);
PrimeRing(R);
AdditiveGroup(R);
MultiplicativeGroup(R);
UnitGroup(R);
Set(R);
IsCommutative(R);
ENDR:=hom< R -> R | > ;
ENDR;
MOL:=hom< R -> ENDR | > ;
MOL;
Residue class ring of integers modulo 8
Residue class ring of integers modulo 8
Ideal of residue class ring of integers modulo 8 generated by 2
Residue class ring of integers modulo 8
8
8
8
[ <2, 3> ]
Residue class ring of integers modulo 8
Abelian Group isomorphic to Z/8
Defined on 1 generator
Relations:
8*$.1 = 0
Abelian Group isomorphic to Z/2 + Z/2
Defined on 2 generators
Relations:
2*$.1 = 0
2*$.2 = 0
Abelian Group isomorphic to Z/2 + Z/2
Defined on 2 generators
Relations:
2*$.1 = 0
2*$.2 = 0
{ 0, 1, 2, 3, 4, 5, 6, 7 }
true
Mapping from: RngIntRes: R to RngIntRes: R
>> MOright:=hom< R -> ENDR | > ;
^[br][br]-=-=-=-=- 以下内容由 cjsh 在 时添加 -=-=-=-=-
Z:=8;
R:=IntegerRing(Z) ;
R;
I3:=ideal< R | 3 > ;
I3;
I10:=ideal< R | 10 > ;
I10;
RR:=ResidueClassRing(Z);
RR;
RC:=Characteristic(R) ;
RC;
# R;
Modulus(R);
FactoredModulus(R);
PrimeRing(R);
AdditiveGroup(R);
MultiplicativeGroup(R);
UnitGroup(R);
Set(R);
IsCommutative(R);
ENDR:=hom< R -> R | > ;
ENDR;
MOL:=hom< R -> ENDR | > ;
MOL;
RRR:=MatrixRing(RealField(12), 3);
RRR;
hhh := hom< RRR -> RRR | >;
hhh;
D := DiagonalMatrix(GF(23), [1, 2, -3]);
RRRR:=MatrixRing(D, 3);
RRRR;
hhhh := hom< RRRR -> RRRR | >;
hhhh;
GLNZ:=RandomGLnZ(10, 6, 1);
GLNZ;
hhhhh := hom< GLNZ -> GLNZ | >;
hhhhh;
Residue class ring of integers modulo 8
Residue class ring of integers modulo 8
Ideal of residue class ring of integers modulo 8 generated by 2
Residue class ring of integers modulo 8
8
8
8
[ <2, 3> ]
Residue class ring of integers modulo 8
Abelian Group isomorphic to Z/8
Defined on 1 generator
Relations:
8*$.1 = 0
Abelian Group isomorphic to Z/2 + Z/2
Defined on 2 generators
Relations:
2*$.1 = 0
2*$.2 = 0
Abelian Group isomorphic to Z/2 + Z/2
Defined on 2 generators
Relations:
2*$.1 = 0
2*$.2 = 0
{ 0, 1, 2, 3, 4, 5, 6, 7 }
true
Mapping from: RngIntRes: R to RngIntRes: R
>> MOL:=hom< R -> ENDR | > ;
^
Runtime error in hom< ... >: Homomorphism has an invalid codomain
>> MOL;
^
User error: Identifier ';MOL'; has not been declared or assigned
Full Matrix Algebra of degree 3 over Real field of precision 12
>> hhh := hom< RRR -> RRR | >;
^
Runtime error in hom< ... >: Rhs arity (0) should be 1
>> hhh;
^
User error: Identifier ';hhh'; has not been declared or assigned
>> RRRR:=MatrixRing(D, 3);
^
Runtime error in ';MatrixRing';: Bad argument types
Argument types given: AlgMatElt[FldFin], RngIntElt
>> RRRR;
^
User error: Identifier ';RRRR'; has not been declared or assigned
>> hhhh := hom< RRRR -> RRRR | >;
^
User error: Identifier ';RRRR'; has not been declared or assigned
>> hhhh;
^
User error: Identifier ';hhhh'; has not been declared or assigned
[ 1 0 0 0 0 0 0 0 0 0]
[ 0 1 0 0 0 0 0 0 0 0]
[ 0 0 1 0 0 0 0 0 0 0]
[ 0 0 0 1 0 0 0 0 0 0]
[ 0 0 0 0 -1 0 0 0 0 0]
[ 0 0 0 0 0 1 0 0 0 0]
[ 0 0 0 0 0 0 1 0 0 0]
[ 0 5 0 0 0 0 0 1 0 0]
[ 0 0 0 0 0 0 0 0 1 0]
[ 0 0 0 0 0 0 0 0 0 1]
>> hhhhh := hom< GLNZ -> GLNZ | >;
^
Runtime error in ';->';: Bad argument types
Argument types given: AlgMatElt[RngInt], AlgMatElt[RngInt]
>> hhhhh;
^
User error: Identifier ';hhhhh'; has not been declared or assigned[br][br]-=-=-=-=- 以下内容由 cjsh 在 时添加 -=-=-=-=-
Z := Integers();
> Q := RationalField();
> P := PolynomialRing(Z);
> S, h := ChangeRing(P, Q);
> h(x^3-2*x+5);
> S ! (x^3-2*x+5);
> m := hom< Z -> Q | x :-> 3*x >;
> S, h := ChangeRing(P, Q, m);
> h(x^3-2*x+5);
mm := hom< Z -> Z | x :-> x >;
mm;
mmm := hom< Q -> Q | x :-> x >;
mmm;
y^3 - 2*y + 5
y^3 - 2*y + 5
3*y^3 - 6*y + 15
Mapping from: RngInt: Z to RngInt: Z given by a rule [no inverse]
Mapping from: FldRat: Q to FldRat: Q given by a rule [no inverse]
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