Construction of Modules of n-tuples
RSpace(R, n) : Rng, RngIntElt -> ModTupRng
RSpace(R, n, F) : Rng, RngIntElt, Mtrx -> ModTupRng
Example ModRng_CreateZ6 (H64E1)
Construction of Modules of m x n Matrices
RMatrixSpace(R, m, n) : Rng, RngIntElt, RngIntElt -> ModMatRng
Construction of a Module with Specified Basis
RModuleWithBasis(Q) : [ModRngElt] -> ModTupRng
RMatrixSpaceWithBasis(Q) : [ModTupRngElt] -> ModMatRng
Accessing Module Information
M . i : ModTupRng, RngIntElt -> ModElt
CoefficientRing(M) : ModTupRng -> Rng
Generators(M) : ModTupRng -> { ModTupElt }
OverDimension(M) : ModTupRng -> RngIntElt
OverDimension(u) : ModTupRngElt -> RngIntElt
Moduli(M) : ModTupRng -> [ RngElt ]
Parent(u) : ModTupElt -> ModRng
Generic(M) : ModRng -> ModRng
Changing the Coefficient Ring
ChangeRing(M, S) : ModRng, Rng -> ModRng, Map
ChangeRing(M, S, f) : ModRng, Rng, Map -> ModRng, Map
Direct Sums
DirectSum(M, N) : ModRng, ModRng -> ModRng, Map, Map, Map, Map
DirectSum(Q) : [ ModRng ] -> [ ModRng ], [ Map ], [ Map ]
Construction of Elements
elt< M | a_1, ..., a_n > : ModTupRng, List -> ModTupRngElt
M ! Q : ModTupRng, [RngElt] -> ModTupRngElt
CharacteristicVector(M, S) : ModRng, { RngIntElt } -> ModRngElt
Zero(M) : ModRng, RngIntElt -> ModRngElt
Random(M) : ModRng -> ModRngElt
Example ModRng_Elements (H64E2)
Deconstruction of Elements
ElementToSequence(u) : ModTupRngElt -> [RngElt]
Arithmetic
u + v : ModTupElt, ModTupElt -> ModTupElt
- u : ModTupElt -> ModTupElt
u - v : ModTupElt, ModTupElt -> ModTupElt
x * u : RngElt, ModTupElt -> ModTupElt
u * x : ModTupElt, RngElt -> ModTupElt
u / x : ModTupElt, RngElt -> ModTupElt
Indexing
u[i] : ModTupRngElt, RngIntElt -> RngElt
u[i] := x : ModTupRngElt, RngIntElt, RngElt -> ModTupRngElt
Normalization
Normalize(u) : ModTupElt -> ModTupElt
Rotate(u, k) : ModTupElt, RngIntElt -> ModTupElt
Rotate(~u, k) : ModTupElt, RngIntElt ->
Example ModRng_Operations (H64E3)
Properties of Vectors
IsZero(u) : ModTupElt -> BoolElt
Depth(v) : ModTupRngElt -> RngIntElt
Support(u) : ModTupRngElt -> { RngElt }
Weight(u) : ModTupRngElt -> RngIntElt
Inner Products
(u, v) : ModTupRngElt, ModTupRngElt -> RngElt
Norm(u) : ModTupRngElt -> RngElt
Bases
Basis(M) : ModTupRng -> [ModTupRngElt]
Rank(M) : ModTupRng -> RngIntElt
Coordinates(M, u) : ModTupRng, ModTupRngElt -> [RngElt]
Construction of Submodules
sub<M | L> : ModTupRng, List -> ModTupRng
Example ModRng_Submodule (H64E4)
Membership and Equality
u in M : ModTupRngElt, ModTupRng -> BoolElt
u notin M : ModTupRngElt, ModTupRng -> BoolElt
N subset M : ModTupRng, ModTupRng -> BoolElt
N notsubset M : ModTupRng, ModTupRng -> BoolElt
N eq M : ModTupRng, ModTupRng -> BoolElt
N ne M : ModTupRng, ModTupRng -> BoolElt
Operations on Submodules
M + N : ModTupRng, ModTupRng -> ModTupRng
M meet N : ModTupRng, ModTupRng -> ModTupRng
Construction of Quotient Modules
quo<M | L> : ModTupRng, List -> ModTupRng
öm_(R)(M, N) for R-modules
Hom(M, N) : ModTupRng, ModTupRng -> ModMatRng
RMatrixSpace(R, m, n) : Rng, RngIntElt, RngIntElt -> ModMatRng
Example ModRng_Create (H64E5)
öm_(R)(M, N) for matrix modules
Hom(M, N, "right") : ModMatRng, ModMatRng, MonStgElt -> ModMatRng
Hom(M, N, "left") : ModMatRng, ModMatRng, MonStgElt -> ModMatRng
Example ModRng_CreateHom (H64E6)
Modules öm_(R)(M, N) with Given Basis
RMatrixSpaceWithBasis(Q) : [ ModMatRngElt ] -> ModMatRng
KMatrixSpaceWithBasis(Q) : [ ModMatRngElt ] -> ModMatRng
The Endomorphsim Ring
EndomorphismAlgebra(M) : ModTupRng -> AlgMat
Example ModRng_CreateHom (H64E7)
The Reduced Form of a Matrix Module
Reduce(H) : ModMatRng -> ModMatRng, Map
Example ModRng_Reduce (H64E8)
Example ModRng_ReduceHom (H64E9)
Construction of a Matrix
M ! Q : ModMatRng, [RngElt] -> ModMatRngElt
Example ModRng_Matrix (H64E10)
Element Operations
u * a : ModTupRngElt, ModMatRngElt -> ModTupRngElt
a * b : ModMatRngElt, ModMatRngElt -> ModMatRngElt
a ^ -1 : ModMatRngElt, RngIntElt -> ModMatRngElt
Codomain(S) : ModMatRng -> ModTupRng
Codomain(a) : ModMatRngElt -> ModTupRng
Cokernel(a) : ModMatRngElt -> ModTupRng
Domain(S) : ModMatRng -> ModTupRng
Domain(a) : ModMatRngElt -> ModTupRng
Image(a) : ModMatRngElt -> ModTupRng
Kernel(a) : ModMatRngElt -> ModTupRng
Morphism(M, N) : ModTupRng, ModTupRng -> ModMatRngElt
Rank(a) : ModMatRngElt -> RngIntElt
IsBijective(a) : ModMatRngElt -> BoolElt
IsInjective(a) : ModMatRngElt -> BoolElt
IsSurjective(a) : ModMatRngElt -> BoolElt