FREE MODULES
Acknowledgements
Introduction
Free Modules
Module Categories
Presentation of Submodules
Notation
Definition of a Module
Construction of Modules of n-tuples
Construction of Modules of m x n Matrices
Construction of a Module with Specified Basis
Accessing Module Information
Standard Constructions
Changing the Coefficient Ring
Direct Sums
Construction of Elements
Deconstruction of Elements
Operations on Module Elements
Arithmetic
Indexing
Normalization
Properties of Vectors
Inner Products
Bases
Submodules
Construction of Submodules
Operations on Submodules
Membership and Equality
Operations on Submodules
Quotient Modules
Construction of Quotient Modules
Homomorphisms
HomR(M, N) for R-modules
HomR(M, N) for Matrix Modules
Modules HomR(M, N) with Given Basis
The Endomorphism Ring
The Reduced Form of a Matrix Module
Construction of a Matrix
Element Operations
Introduction
Free Modules
Module Categories
Presentation of Submodules
Notation
Definition of a Module
Construction of Modules of n-tuples
RSpace(R, n) : Rng, RngIntElt -> ModTupRng
RSpace(R, n, F) : Rng, RngIntElt, Mtrx -> ModTupRng
Example ModRng_CreateZ6 (H60E1)
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) : [ModFldElt] -> ModFld
RMatrixSpaceWithBasis(Q) : [ModTupRngElt] -> ModMatRng
Accessing Module Information
M . i : ModTupRng, RngIntElt -> ModElt
CoefficientRing(M) : ModTupRng -> Rng
Generators(M) : ModTupRng -> { ModTupRngElt }
OverDimension(M) : ModTupRng -> RngIntElt
OverDimension(u) : ModTupRngElt -> RngIntElt
Moduli(M) : ModTupRng -> [ RngElt ]
Parent(u) : ModTupRngElt -> ModRng
Generic(M) : ModRng -> ModRng
Standard Constructions
Changing the Coefficient Ring
ChangeRing(M, S) : ModRng, Rng -> ModRng, Map
ChangeRing(M, S, f) : ModRng, Rng, Map -> ModRng, Map
ChangeUniverse(~x, R) : ModTupRngElt, Rng -> 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 | a1, ..., an > : ModTupRng, List -> ModTupRngElt
M ! Q : ModTupRng, [RngElt] -> ModTupRngElt
CharacteristicVector(M, S) : ModRng, { RngIntElt } -> ModRngElt
Zero(M) : ModRng -> ModRngElt
Random(M) : ModRng -> ModRngElt
Example ModRng_Elements (H60E2)
Deconstruction of Elements
ElementToSequence(u) : ModTupRngElt -> [RngElt]
Operations on Module Elements
Arithmetic
u + v : ModTupRngElt, ModTupRngElt -> ModTupRngElt
- u : ModTupRngElt -> ModTupRngElt
u - v : ModTupRngElt, ModTupRngElt -> ModTupRngElt
x * u : RngElt, ModTupRngElt -> ModTupRngElt
u * x : ModTupRngElt, RngElt -> ModTupRngElt
u / x : ModTupRngElt, RngElt -> ModTupRngElt
Indexing
u[i] : ModTupRngElt, RngIntElt -> RngElt
u[i] := x : ModTupRngElt, RngIntElt, RngElt -> ModTupRngElt
Normalization
Normalize(u) : ModTupRngElt -> ModTupRngElt
Rotate(u, k) : ModTupRngElt, RngIntElt -> ModTupRngElt
Rotate(~u, k) : ModTupRngElt, RngIntElt ->
Example ModRng_Operations (H60E3)
Properties of Vectors
IsZero(u) : ModTupRngElt -> 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]
Submodules
Construction of Submodules
sub<M | L> : ModTupRng, List -> ModTupRng
Example ModRng_Submodule (H60E4)
Operations on Submodules
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
Quotient Modules
Construction of Quotient Modules
quo<M | L> : ModTupRng, List -> ModTupRng
Homomorphisms
HomR(M, N) for R-modules
Hom(M, N) : ModTupRng, ModTupRng -> ModMatRng
RMatrixSpace(R, m, n) : Rng, RngIntElt, RngIntElt -> ModMatRng
Example ModRng_Create (H60E5)
HomR(M, N) for Matrix Modules
Hom(M, N, "right") : ModMatRng, ModMatRng, MonStgElt -> ModMatRng
Hom(M, N, "left") : ModMatRng, ModMatRng, MonStgElt -> ModMatRng
Example ModRng_CreateHom (H60E6)
Modules HomR(M, N) with Given Basis
RMatrixSpaceWithBasis(Q) : [ ModMatRngElt ] -> ModMatRng
KMatrixSpaceWithBasis(Q) : [ ModMatRngElt ] -> ModMatRng
The Endomorphism Ring
EndomorphismAlgebra(M) : ModTupRng -> AlgMat
Example ModRng_CreateHom (H60E7)
The Reduced Form of a Matrix Module
Reduce(H) : ModMatRng -> ModMatRng, Map
Example ModRng_Reduce (H60E8)
Example ModRng_ReduceHom (H60E9)
Construction of a Matrix
M ! Q : ModMatRng, [RngElt] -> ModMatRngElt
Example ModRng_Matrix (H60E10)
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
Example ModRng_Operations (H60E11)
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