RING反同态和左模例子

cjsh

[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]