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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
 
Elements
 
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 Endomorphsim Ring
      The Reduced Form of a Matrix Module
      Construction of a Matrix
      Element Operations







DETAILS

 
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 (H54E1)

      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 ]

 
Elements

 
Construction of Elements
      elt< M | a1, ..., an > : 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 (H54E2)

      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 (H54E3)

      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 (H54E4)

      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 (H54E5)

      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 (H54E6)

      Modules HomR(M, N) with Given Basis
            RMatrixSpaceWithBasis(Q) : [ ModMatRngElt ] -> ModMatRng
            KMatrixSpaceWithBasis(Q) : [ ModMatRngElt ] -> ModMatRng

      The Endomorphsim Ring
            EndomorphismAlgebra(M) : ModTupRng -> AlgMat
            Example ModRng_CreateHom (H54E7)

      The Reduced Form of a Matrix Module
            Reduce(H) : ModMatRng -> ModMatRng, Map
            Example ModRng_Reduce (H54E8)
            Example ModRng_ReduceHom (H54E9)

      Construction of a Matrix
            M ! Q : ModMatRng, [RngElt] -> ModMatRngElt
            Example ModRng_Matrix (H54E10)

      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 (H54E11)

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Version: V2.19 of Wed Apr 24 15:09:57 EST 2013