# Free Modules - [Introduction](introduction.md) - [Free Modules](introduction.md#free-modules) - [Module Categories](introduction.md#module-categories) - [Presentation of Submodules](introduction.md#presentation-of-submodules) - [Notation](introduction.md#notation) - [Definition of a Module](creation-module.md) - [Construction of Modules of $n$-tuples](creation-module.md#construction-of-modules-of-n-tuples) - [`RSpace(R, n): Rng, RngIntElt → ModTupRng`](creation-module.md#function-modrng-rspace) - [`RModule(R, n): Rng, RngIntElt → ModRng`](creation-module.md#function-modrng-rmodule) - [`RSpace(R, n, F): Rng, RngIntElt, Mtrx → ModTupRng`](creation-module.md#function-rspace-rng-rngintelt-mtrx) - [`Example: Create Z6`](creation-module.md#example-ex-d42492) - [Construction of Modules of $m \times n$ Matrices](creation-module.md#construction-of-modules-of-m-times-n-matrices) - [`RMatrixSpace(R, m, n): Rng, RngIntElt, RngIntElt → ModMatRng`](creation-module.md#function-rmatrixspace-rng-rngintelt-rngintelt) - [Construction of a Module with Specified Basis](creation-module.md#construction-of-a-module-with-specified-basis) - [`RModuleWithBasis(Q): [ModFldElt] → ModFld`](creation-module.md#function-rmodulewithbasis-modfldelt) - [`RSpaceWithBasis(Q): [ModTupRngElt] → ModTupRng`](creation-module.md#function-rspacewithbasis-modtuprngelt) - [`RSpaceWithBasis(a): AlgMatElt → ModTupRng`](creation-module.md#function-rspacewithbasis-algmatelt) - [`RSpaceWithBasis(a): ModMatRngElt → ModTupRng`](creation-module.md#function-rspacewithbasis-modmatrngelt) - [`RMatrixSpaceWithBasis(Q): [ModTupRngElt] → ModMatRng`](creation-module.md#function-rmatrixspacewithbasis-modtuprngelt) - [Accessing Module Information](access.md) - [`M . i: ModTupRng, RngIntElt → ModElt`](access.md#operation-operation-modtuprng-rngintelt-modelt) - [`CoefficientRing(M): ModTupRng → Rng`](access.md#function-coefficientring-modtuprng) - [`BaseRing(M): ModTupRng → Rng`](access.md#function-basering-modtuprng) - [`CoefficientRing(M): ModRng → Rng`](access.md#function-coefficientring-modrng) - [`BaseRing(M): ModRng → Rng`](access.md#function-basering-modrng) - [`CoefficientField(M): ModFld → Fld`](access.md#function-coefficientfield-modfld) - [`BaseField(M): ModFld → Fld`](access.md#function-basefield-modfld) - [`Generators(M): ModTupRng → { ModTupRngElt }`](access.md#function-generators-modtuprng) - [`OverDimension(M): ModTupRng → RngIntElt`](access.md#function-overdimension-modtuprng) - [`OverDimension(u): ModTupRngElt → RngIntElt`](access.md#function-overdimension-modtuprngelt) - [`Moduli(M): ModTupRng → [ RngElt ]`](access.md#function-moduli-modtuprng) - [`Parent(u): ModTupRngElt → ModRng`](access.md#function-parent-modtuprngelt) - [`Generic(M): ModRng → ModRng`](access.md#function-generic-modrng) - [Standard Constructions](operation.md) - [Changing the Coefficient Ring](operation.md#changing-the-coefficient-ring) - [`ChangeRing(M, S): ModRng, Rng → ModRng, Map`](operation.md#function-changering-modrng-rng) - [`ChangeRing(M, S, f): ModRng, Rng, Map → ModRng, Map`](operation.md#function-changering-modrng-rng-map) - [`ChangeUniverse(~x, R): ModTupRngElt, Rng → ModRng, Map`](operation.md#function-changeuniverse-modtuprngelt-rng-ref) - [Direct Sums](operation.md#direct-sums) - [`DirectSum(M, N): ModRng, ModRng → ModRng, Map, Map, Map, Map`](operation.md#function-directsum-modrng-modrng) - [`DirectSum(Q): [ ModRng ] → [ ModRng ], [ Map ], [ Map ]`](operation.md#function-directsum-modrng) - [Construction of Elements](creation.md) - [`elt< M | a₁, ..., aₙ >: ModTupRng, List → ModTupRngElt`](creation.md#constructor-constructor-elt-modtuprng-list-modtuprngelt) - [`M ! Q: ModTupRng, [RngElt] → ModTupRngElt`](creation.md#operation-op-modtuprng-rngelt) - [`CharacteristicVector(M, S): ModRng, { RngIntElt } → ModRngElt`](creation.md#function-characteristicvector-modrng-rngintelt) - [`Zero(M): ModRng → ModRngElt`](creation.md#function-zero-modrng) - [`M ! 0: ModRng, RngIntElt → ModRngElt`](creation.md#operation-op-modrng-rngintelt) - [`Random(M): ModRng → ModRngElt`](creation.md#function-random-modrng) - [`Example: Elements`](creation.md#example-ex-cd5034) - [Deconstruction of Elements](creation.md#deconstruction-of-elements) - [`ElementToSequence(u): ModTupRngElt → [RngElt]`](creation.md#function-elementtosequence-modtuprngelt) - [`Eltseq(u): ModTupRngElt → [RngElt]`](creation.md#function-eltseq-modtuprngelt) - [Operations on Module Elements](creation.md#operations-on-module-elements) - [Arithmetic](creation.md#arithmetic) - [`u + v: ModTupRngElt, ModTupRngElt → ModTupRngElt`](creation.md#operation-op-plus-modtuprngelt-modtuprngelt) - [`- u: ModTupRngElt → ModTupRngElt`](creation.md#operation-operation-modtuprngelt-modtuprngelt) - [`u - v: ModTupRngElt, ModTupRngElt → ModTupRngElt`](creation.md#operation-op-minus-modtuprngelt-modtuprngelt) - [`x * u: RngElt, ModTupRngElt → ModTupRngElt`](creation.md#operation-op-times-rngelt-modtuprngelt) - [`u * x: ModTupRngElt, RngElt → ModTupRngElt`](creation.md#operation-op-times-modtuprngelt-rngelt) - [`u / x: ModTupRngElt, RngElt → ModTupRngElt`](creation.md#operation-op-div-modtuprngelt-rngelt) - [Indexing](creation.md#indexing) - [`u[i]: ModTupRngElt, RngIntElt → RngElt`](creation.md#indexed-indexed-u-modtuprngelt-rngintelt-rngelt) - [`u[i] := x: ModTupRngElt, RngIntElt, RngElt → ModTupRngElt`](creation.md#literal-literal-u-i-x-modtuprngelt-rngintelt-rngelt-modtuprngelt) - [Normalization](creation.md#normalization) - [`Normalize(u): ModTupRngElt → ModTupRngElt`](creation.md#function-normalize-modtuprngelt) - [`Normalise(u): ModTupRngElt → ModTupRngElt`](creation.md#function-normalise-modtuprngelt) - [`Rotate(u, k): ModTupRngElt, RngIntElt → ModTupRngElt`](creation.md#function-rotate-modtuprngelt-rngintelt) - [`Rotate(~u, k): ModTupRngElt, RngIntElt`](creation.md#function-rotate-modtuprngelt-rngintelt-ref) - [`Example: Operations`](creation.md#example-ex-8a5f2c) - [Properties of Vectors](creation.md#properties-of-vectors) - [`IsZero(u): ModTupRngElt → BoolElt`](creation.md#function-iszero-modtuprngelt) - [`Depth(v): ModTupRngElt → RngIntElt`](creation.md#function-depth-modtuprngelt) - [`Support(u): ModTupRngElt → { RngElt }`](creation.md#function-support-modtuprngelt) - [`Weight(u): ModTupRngElt → RngIntElt`](creation.md#function-weight-modtuprngelt) - [Inner Products](creation.md#inner-products) - [`(u, v): ModTupRngElt, ModTupRngElt → RngElt`](creation.md#literal-literal-u-v-modtuprngelt-modtuprngelt-rngelt) - [`InnerProduct(u, v): ModTupRngElt, ModTupRngElt → RngElt`](creation.md#function-innerproduct-modtuprngelt-modtuprngelt) - [`Norm(u): ModTupRngElt → RngElt`](creation.md#function-norm-modtuprngelt) - [Bases](basis.md) - [`Basis(M): ModTupRng → [ModTupRngElt]`](basis.md#function-basis-modtuprng) - [`Rank(M): ModTupRng → RngIntElt`](basis.md#function-rank-modtuprng) - [`Coordinates(M, u): ModTupRng, ModTupRngElt → [RngElt]`](basis.md#function-coordinates-modtuprng-modtuprngelt) - [Submodules](submodule.md) - [Construction of Submodules](submodule.md#construction-of-submodules) - [`sub: ModTupRng, List → ModTupRng`](submodule.md#constructor-constructor-sub-modtuprng-list-modtuprng) - [`sub: ModMatRng, List → ModMatRng`](submodule.md#constructor-constructor-sub-modmatrng-list-modmatrng) - [`Example: Submodule`](submodule.md#example-ex-e4a021) - [Operations on Submodules](submodule.md#operations-on-submodules) - [Membership and Equality](submodule.md#membership-and-equality) - [`u in M: ModTupRngElt, ModTupRng → BoolElt`](submodule.md#operation-op-in-modtuprngelt-modtuprng) - [`u in M: ModMatRngElt, ModMatRng → BoolElt`](submodule.md#operation-op-in-modmatrngelt-modmatrng) - [`u notin M: ModTupRngElt, ModTupRng → BoolElt`](submodule.md#operation-op-notin-modtuprngelt-modtuprng) - [`u notin M: ModMatRngElt, ModMatRng → BoolElt`](submodule.md#operation-op-notin-modmatrngelt-modmatrng) - [`N subset M: ModTupRng, ModTupRng → BoolElt`](submodule.md#operation-op-subset-modtuprng-modtuprng) - [`N subset M: ModMatRng, ModMatRng → BoolElt`](submodule.md#operation-op-subset-modmatrng-modmatrng) - [`N notsubset M: ModTupRng, ModTupRng → BoolElt`](submodule.md#operation-operation-notsubset-modtuprng-modtuprng-boolelt) - [`N notsubset M: ModMatRng, ModMatRng → BoolElt`](submodule.md#operation-operation-notsubset-modmatrng-modmatrng-boolelt) - [`N eq M: ModTupRng, ModTupRng → BoolElt`](submodule.md#operation-op-eq-modtuprng-modtuprng) - [`M eq N: ModMatRng, ModMatRng → BoolElt`](submodule.md#operation-op-eq-modmatrng-modmatrng) - [`N ne M: ModTupRng, ModTupRng → BoolElt`](submodule.md#operation-op-ne-modtuprng-modtuprng) - [`M ne N: ModMatRng, ModMatRng → BoolElt`](submodule.md#operation-op-ne-modmatrng-modmatrng) - [Operations on Submodules](submodule.md#id1) - [`M + N: ModTupRng, ModTupRng → ModTupRng`](submodule.md#operation-op-plus-modtuprng-modtuprng) - [`M + N: ModMatRng, ModMatRng → ModMatRng`](submodule.md#operation-op-plus-modmatrng-modmatrng) - [`M meet N: ModTupRng, ModTupRng → ModTupRng`](submodule.md#operation-op-meet-modtuprng-modtuprng) - [`M meet N: ModMatRng, ModMatRng → ModMatRng`](submodule.md#operation-op-meet-modmatrng-modmatrng) - [Quotient Modules](quotient.md) - [Construction of Quotient Modules](quotient.md#construction-of-quotient-modules) - [`quo: ModTupRng, List → ModTupRng`](quotient.md#constructor-constructor-quo-modtuprng-list-modtuprng) - [`quo: ModMatRng, List → ModMatRng`](quotient.md#constructor-constructor-quo-modmatrng-list-modmatrng) - [Homomorphisms](homomorphisms.md) - [${\operatorname{Hom}}_{R}(M, N)$ for $R$-modules](homomorphisms.md#operatorname-hom-r-m-n-for-r-modules) - [`Hom(M, N): ModTupRng, ModTupRng → ModMatRng`](homomorphisms.md#function-hom-modtuprng-modtuprng) - [`RMatrixSpace(R, m, n): Rng, RngIntElt, RngIntElt → ModMatRng`](homomorphisms.md#function-rmatrixspace-rng-rngintelt-rngintelt-2) - [`Example: Create`](homomorphisms.md#example-ex-bf6e98) - [${\operatorname{Hom}}_{R}(M, N)$ for Matrix Modules](homomorphisms.md#operatorname-hom-r-m-n-for-matrix-modules) - [`Hom(M, N, "right"): ModMatRng, ModMatRng, MonStgElt → ModMatRng`](homomorphisms.md#function-hom-modmatrng-modmatrng-monstgelt) - [`Hom(M, N, "left"): ModMatRng, ModMatRng, MonStgElt → ModMatRng`](homomorphisms.md#function-hom-modmatrng-modmatrng-monstgelt-2) - [`Example: Create Hom`](homomorphisms.md#example-ex-e3a29e) - [Modules ${\operatorname{Hom}}_{R}(M, N)$ with Given Basis](homomorphisms.md#modules-operatorname-hom-r-m-n-with-given-basis) - [`RMatrixSpaceWithBasis(Q): [ ModMatRngElt ] → ModMatRng`](homomorphisms.md#function-rmatrixspacewithbasis-modmatrngelt) - [`KMatrixSpaceWithBasis(Q): [ ModMatRngElt ] → ModMatRng`](homomorphisms.md#function-kmatrixspacewithbasis-modmatrngelt) - [The Endomorphism Ring](homomorphisms.md#the-endomorphism-ring) - [`EndomorphismAlgebra(M): ModTupRng → AlgMat`](homomorphisms.md#function-endomorphismalgebra-modtuprng) - [`Example: Create Hom`](homomorphisms.md#example-ex-7acde1) - [The Reduced Form of a Matrix Module](homomorphisms.md#the-reduced-form-of-a-matrix-module) - [`Reduce(H): ModMatRng → ModMatRng, Map`](homomorphisms.md#function-reduce-modmatrng) - [`Example: Reduce`](homomorphisms.md#example-ex-ecaba6) - [`Example: Reduce Hom`](homomorphisms.md#example-ex-4c472d) - [Construction of a Matrix](homomorphisms.md#construction-of-a-matrix) - [`M ! Q: ModMatRng, [RngElt] → ModMatRngElt`](homomorphisms.md#operation-op-modmatrng-rngelt) - [`Example: Matrix`](homomorphisms.md#example-ex-9b8959) - [Element Operations](homomorphisms.md#element-operations) - [`u * a: ModTupRngElt, ModMatRngElt → ModTupRngElt`](homomorphisms.md#operation-op-times-modtuprngelt-modmatrngelt) - [`a(u): ModTupRngElt, ModMatRngElt → ModTupRngElt`](homomorphisms.md#literal-a-modtuprngelt-modmatrngelt) - [`a * b: ModMatRngElt, ModMatRngElt → ModMatRngElt`](homomorphisms.md#operation-op-times-modmatrngelt-modmatrngelt) - [`a ^ -1: ModMatRngElt, RngIntElt → ModMatRngElt`](homomorphisms.md#operation-op-pow-modmatrngelt-rngintelt) - [`Codomain(S): ModMatRng → ModTupRng`](homomorphisms.md#function-codomain-modmatrng) - [`Codomain(a): ModMatRngElt → ModTupRng`](homomorphisms.md#function-codomain-modmatrngelt) - [`Cokernel(a): ModMatRngElt → ModTupRng`](homomorphisms.md#function-cokernel-modmatrngelt) - [`Domain(S): ModMatRng → ModTupRng`](homomorphisms.md#function-domain-modmatrng) - [`Domain(a): ModMatRngElt → ModTupRng`](homomorphisms.md#function-domain-modmatrngelt) - [`Image(a): ModMatRngElt → ModTupRng`](homomorphisms.md#function-image-modmatrngelt) - [`Kernel(a): ModMatRngElt → ModTupRng`](homomorphisms.md#function-kernel-modmatrngelt) - [`NullSpace(a): ModMatRngElt → ModTupRng`](homomorphisms.md#function-nullspace-modmatrngelt) - [`Morphism(M, N): ModTupRng, ModTupRng → ModMatRngElt`](homomorphisms.md#function-morphism-modtuprng-modtuprng) - [`Rank(a): ModMatRngElt → RngIntElt`](homomorphisms.md#function-rank-modmatrngelt) - [`IsBijective(a): ModMatRngElt → BoolElt`](homomorphisms.md#function-isbijective-modmatrngelt) - [`IsInjective(a): ModMatRngElt → BoolElt`](homomorphisms.md#function-isinjective-modmatrngelt) - [`IsSurjective(a): ModMatRngElt → BoolElt`](homomorphisms.md#function-issurjective-modmatrngelt) - [`Example: Operations`](homomorphisms.md#example-ex-85af3b)