# Associative Algebras

- [Introduction](introduction.md)

- [Construction of Associative Algebras](creation.md)

  - [Construction of an Associative Structure Constant Algebra](creation.md#construction-of-an-associative-structure-constant-algebra)

    - [`AssociativeAlgebra< R, n | Q : parameters >: Rng, RngIntElt, SeqEnum → AlgAss`](creation.md#constructor-constructor-associativealgebra-rng-rngintelt-seqenum-algass)

    - [`AssociativeAlgebra< M | Q : parameters >: ModTupRng, SeqEnum → AlgAss`](creation.md#constructor-constructor-associativealgebra-modtuprng-seqenum-algass)

    - [`AssociativeAlgebra< R, n | T : parameters >: Rng, RngIntElt, SeqEnum → AlgAss`](creation.md#constructor-constructor-associativealgebra-rng-rngintelt-seqenum-algass-2)

    - [`AssociativeAlgebra(A): AlgGen → AlgAss`](creation.md#function-associativealgebra-alggen)

    - [`ChangeBasis(A, B): AlgAss, {[AlgAssElt]} → AlgAss`](creation.md#function-changebasis-algass-algasselt)

    - [`ChangeBasis(A, B): AlgAss, {[ModTupFldElt]} → AlgAss`](creation.md#function-changebasis-algass-modtupfldelt)

    - [`ChangeBasis(A, B): AlgAss, Mtrx → AlgAss`](creation.md#function-changebasis-algass-mtrx)

  - [Associative Structure Constant Algebras from other Algebras](creation.md#associative-structure-constant-algebras-from-other-algebras)

    - [`Algebra(A): AlgGrp → AlgAss, Map`](creation.md#function-algebra-alggrp)

    - [`Algebra(F, E): FldFin, FldFin → AlgAss, Map;`](creation.md#function-algebra-fldfin-fldfin)

    - [`Algebra(F, E): FldNum, FldNum → AlgAss, Map;`](creation.md#function-algebra-fldnum-fldnum)

    - [`AlgebraOverCenter(A): Alg → AlgAss, Map;`](creation.md#function-algebraovercenter-alg)

- [Operations on Algebras and their Elements](ops.md)

  - [Operations on Algebras](ops.md#operations-on-algebras)

    - [`Centre(A): AlgAss → AlgAss`](ops.md#function-centre-algass)

    - [`Centralizer(A, S): AlgAss, AlgAss → AlgAss`](ops.md#function-centralizer-algass-algass)

    - [`Centraliser(A, S): AlgAss, AlgAss → AlgAss`](ops.md#function-centraliser-algass-algass)

    - [`Idealizer(A, B: parameters): AlgAss, AlgAss → AlgAss`](ops.md#function-idealizer-algass-algass)

    - [`Idealiser(A, B: parameters): AlgAss, AlgAss → AlgAss`](ops.md#function-idealiser-algass-algass)

    - [`LieAlgebra(A): AlgAss → AlgGen, Map`](ops.md#function-liealgebra-algass)

    - [`CommutatorModule(A, B): AlgAss, AlgAss → ModTupRng`](ops.md#function-commutatormodule-algass-algass)

    - [`CommutatorIdeal(A, B): AlgAss, AlgAss → AlgAss`](ops.md#function-commutatorideal-algass-algass)

    - [`LeftAnnihilator(A, B): AlgAss, AlgAss → AlgAss, AlgAss`](ops.md#function-leftannihilator-algass-algass)

    - [`RightAnnihilator(A, B): AlgAss, AlgAss → AlgAss, AlgAss`](ops.md#function-rightannihilator-algass-algass)

    - [`Example: liealg`](ops.md#example-ex-28076e)

    - [`RestrictionOfScalars(A): AlgAss[FldAlg] → AlgAss, Map`](ops.md#function-restrictionofscalars)

    - [`RestrictionOfScalars(A): AlgAss[FldFun] → AlgAss, Map`](ops.md#function-restrictionofscalars-algass-fldfun)

    - [`RestrictionOfScalars(A, F): AlgAss[FldAlg], Fld → AlgAss, Map`](ops.md#function-restrictionofscalars-algass-fldalg-fld)

    - [`RestrictionOfScalars(A, F): AlgAss[FldFun], FldFunG → AlgAss, Map`](ops.md#function-restrictionofscalars-algass-fldfun-fldfung)

    - [`Example: restrict`](ops.md#example-ex-661f6a)

  - [Operations on Elements](ops.md#operations-on-elements)

    - [`Centralizer(A, s): AlgAss, AlgAssElt → AlgAss`](ops.md#function-centralizer-algass-algasselt)

    - [`Centraliser(A, s): AlgAss, AlgAssElt → AlgAss`](ops.md#function-centraliser-algass-algasselt)

    - [`LieBracket(a, b): AlgAssElt, AlgAssElt → AlgAssElt`](ops.md#function-liebracket-algasselt-algasselt)

    - [`(a, b): AlgAssElt, AlgAssElt → AlgAssElt`](ops.md#literal-literal-a-b-algasselt-algasselt-algasselt)

    - [`IsScalar(a): AlgAssElt → BoolElt, RngElt`](ops.md#function-isscalar-algasselt)

    - [`RepresentationMatrix(a, M : parameters): AlgAssElt, AlgAss → AlgMatElt`](ops.md#function-representationmatrix-algasselt-algass)

  - [Representations](ops.md#representations)

    - [`MatrixAlgebra(A): AlgAss → AlgMat`](ops.md#function-matrixalgebra-algass)

    - [`MatrixAlgebra(A, M : parameters): AlgAss, AlgAss → AlgMat, Map`](ops.md#function-matrixalgebra-algass-algass)

    - [`RegularRepresentation(A : parameters): AlgAss → AlgMat, Map`](ops.md#function-regularrepresentation-algass)

  - [Decomposition of an Algebra](ops.md#decomposition-of-an-algebra)

    - [`JacobsonRadical(A): AlgAssV → AlgAssV`](ops.md#function-jacobsonradical-algassv)

    - [`Example: Jac Rad`](ops.md#example-ex-564d9d)

    - [`DirectSumDecomposition(A): AlgAssV → [ AlgAssV ], [ AlgAssVElt ]`](ops.md#function-directsumdecomposition-algassv)

    - [`IndecomposableSummands(A): AlgAssV → [ AlgAssV ], [ AlgAssVElt ]`](ops.md#function-indecomposablesummands-algassv)

    - [`CentralIdempotents(A): AlgAssV → SeqEnum, SeqEnum`](ops.md#function-centralidempotents-algassv)

    - [`Example: Id Pots`](ops.md#example-ex-4239e7)

- [Orders](alg_ass_ord.md)

  - [Construction of Orders](alg_ass_ord.md#construction-of-orders)

    - [`Order(R, S): Rng, SeqEnum[AlgAssVElt] → AlgAssVOrd`](alg_ass_ord.md#function-order-rng-seqenum-algassvelt)

    - [`Order(R, S): RngFunOrd, SeqEnum[AlgAssVElt[FldFunG]] → AlgAssVOrd`](alg_ass_ord.md#function-order-rngfunord-seqenum-algassvelt-fldfung)

    - [`Order(S): SeqEnum[AlgAssVElt[FldAlg]] → AlgAssVOrd`](alg_ass_ord.md#function-order-seqenum-algassvelt-fldalg)

    - [`Order(S): SeqEnum[AlgAssVElt[FldFun]] → AlgAssVOrd`](alg_ass_ord.md#function-order-seqenum-algassvelt-fldfun)

    - [`Order(S, I): SeqEnum[AlgAssVElt[FldAlg]], SeqEnum[RngOrdFracIdl] → AlgAssVOrd`](alg_ass_ord.md#function-order-seqenum-algassvelt-fldalg-seqenum-rngordfracidl)

    - [`Order(S, I): SeqEnum[AlgAssVElt[FldFunG]], SeqEnum[RngFunOrdIdl] → AlgAssVOrd`](alg_ass_ord.md#function-order-seqenum-algassvelt-fldfung-seqenum-rngfunordidl)

    - [`Order(A, m, I): AlgAssV[FldOrd], AlgMatElt[FldOrd], SeqEnum[RngOrdFracIdl] → AlgAssVOrd`](alg_ass_ord.md#function-order-algassv-fldord-algmatelt-fldord-seqenum-rngordfracidl)

    - [`Order(A, m, I): AlgAssV[FldNum], AlgMatElt[FldNum], SeqEnum[RngOrdFracIdl] → AlgAssVOrd, Map`](alg_ass_ord.md#function-order-algassv-fldnum-algmatelt-fldnum-seqenum-rngordfracidl)

    - [`Order(A, m, I): AlgAssV[FldFun], AlgMatElt[FldFun], SeqEnum[RngFunOrdIdl] → AlgAssVOrd`](alg_ass_ord.md#function-order-algassv-fldfun-algmatelt-fldfun-seqenum-rngfunordidl)

    - [`Order(A, pm): AlgAssV[FldOrd], PMat → AlgAssVOrd`](alg_ass_ord.md#function-order-algassv-fldord-pmat)

    - [`Order(A, pm): AlgAssV[FldNum], PMat → AlgAssVOrd, Map`](alg_ass_ord.md#function-order-algassv-fldnum-pmat)

    - [`Order(A, pm): AlgAssV[FldFun], PMat → AlgAssVOrd`](alg_ass_ord.md#function-order-algassv-fldfun-pmat)

    - [`Example: Ord Creat Cyc`](alg_ass_ord.md#example-ex-5ab68c)

    - [`Example: Ord Creat Cyc`](alg_ass_ord.md#example-ex-1d14f4)

    - [`MaximalOrder(A): AlgAssV[FldRat] → AlgAssVOrd`](alg_ass_ord.md#function-maximalorder-algassv-fldrat)

    - [`MaximalOrder(A): AlgAssV[FldAlg] → AlgAssVOrd`](alg_ass_ord.md#function-maximalorder-algassv-fldalg)

    - [`MaximalOrder(O): AlgAssVOrd[RngInt] → AlgAssVOrd`](alg_ass_ord.md#function-maximalorder-algassvord-rngint)

    - [`MaximalOrder(O): AlgAssVOrd[RngOrd] → AlgAssVOrd`](alg_ass_ord.md#function-maximalorder-algassvord-rngord)

    - [`MaximalOrderFinite(A): AlgAssV[FldFunRat] → AlgAssVOrd`](alg_ass_ord.md#function-maximalorderfinite-algassv-fldfunrat)

    - [`MaximalOrderInfinite(A): AlgAssV[FldFunRat] → AlgAssVOrd`](alg_ass_ord.md#function-maximalorderinfinite-algassv-fldfunrat)

    - [`MaximalOrderFinite(A): AlgAssV[FldFun] → AlgAssVOrd`](alg_ass_ord.md#function-maximalorderfinite-algassv-fldfun)

    - [`MaximalOrderInfinite(A): AlgAssV[FldFun] → AlgAssVOrd`](alg_ass_ord.md#function-maximalorderinfinite-algassv-fldfun)

    - [`MaximalOrder(O): AlgAssVOrd[RngUPol] → AlgAssVOrd`](alg_ass_ord.md#function-maximalorder-algassvord-rngupol)

    - [`MaximalOrder(O): AlgAssVOrd[RngVal] → AlgAssVOrd`](alg_ass_ord.md#function-maximalorder-algassvord-rngval)

    - [`MaximalOrder(O): AlgAssVOrd[RngFunOrd] → AlgAssVOrd`](alg_ass_ord.md#function-maximalorder-algassvord-rngfunord)

    - [`Example: Max Ord`](alg_ass_ord.md#example-ex-9fdfa9)

    - [`Example: Max Ord Alg`](alg_ass_ord.md#example-ex-23b948)

    - [`RestrictionOfScalars(O): AlgAssVOrd[RngOrd] → AlgAssVOrd, Map`](alg_ass_ord.md#function-restrictionofscalars-algassvord-rngord)

    - [`RestrictionOfScalars(O): AlgAssVOrd[RngFunOrd] → AlgAssVOrd, Map`](alg_ass_ord.md#function-restrictionofscalars-algassvord-rngfunord)

    - [`RestrictionOfScalars(O, C): AlgAssVOrd[RngOrd], Rng → AlgAssVOrd, Map`](alg_ass_ord.md#function-restrictionofscalars-algassvord-rngord-rng)

    - [`RestrictionOfScalars(O, C): AlgAssVOrd[RngFunOrd], Rng → AlgAssVOrd, Map`](alg_ass_ord.md#function-restrictionofscalars-algassvord-rngfunord-rng)

    - [`Example: restrict`](alg_ass_ord.md#example-ex-bbfaa2)

  - [Attributes](alg_ass_ord.md#attributes)

    - [`BaseRing(O): AlgAssVOrd → Rng`](alg_ass_ord.md#function-basering-algassvord)

    - [`CoefficientRing(O): AlgAssVOrd → Rng`](alg_ass_ord.md#function-coefficientring-algassvord)

    - [`Algebra(O): AlgAssVOrd → AlgAssV`](alg_ass_ord.md#function-algebra-algassvord)

    - [`Degree(O): AlgAssVOrd → RngIntElt`](alg_ass_ord.md#function-degree-algassvord)

    - [`Dimension(O): AlgAssVOrd → RngIntElt`](alg_ass_ord.md#function-dimension-algassvord)

    - [`Discriminant(O): AlgAssVOrd[RngOrd] → RngOrdIdl`](alg_ass_ord.md#function-discriminant-algassvord-rngord)

    - [`Discriminant(O): AlgAssVOrd[RngFunOrd] → RngOrdIdl`](alg_ass_ord.md#function-discriminant-algassvord-rngfunord)

    - [`Discriminant(O): AlgAssVOrd[RngInt] → RngIntElt`](alg_ass_ord.md#function-discriminant-algassvord-rngint)

    - [`Discriminant(O): AlgAssVOrd[RngUPol] → RngIntElt`](alg_ass_ord.md#function-discriminant-algassvord-rngupol)

    - [`Discriminant(O): AlgAssVOrd[RngVal] → RngIntElt`](alg_ass_ord.md#function-discriminant-algassvord-rngval)

    - [`FactoredDiscriminant(O): AlgAssVOrd[RngOrd] → [Tup]`](alg_ass_ord.md#function-factoreddiscriminant-algassvord-rngord)

    - [`MultiplicationTable(O): AlgAssVOrd → SeqEnum`](alg_ass_ord.md#function-multiplicationtable-algassvord)

    - [`Module(O): AlgAssVOrd[RngOrd] → PMat`](alg_ass_ord.md#function-module-algassvord-rngord)

    - [`TraceZeroSubspace(O): AlgAssVOrd → SeqEnum`](alg_ass_ord.md#function-tracezerosubspace-algassvord)

  - [Bases of Orders](alg_ass_ord.md#bases-of-orders)

    - [`PseudoBasis(O): AlgAssVOrd[RngOrd] → SeqEnum`](alg_ass_ord.md#function-pseudobasis-algassvord-rngord)

    - [`PseudoBasis(O): AlgAssVOrd[RngFunOrd] → SeqEnum`](alg_ass_ord.md#function-pseudobasis-algassvord-rngfunord)

    - [`PseudoMatrix(O): AlgAssVOrd[RngOrd]> → PMat`](alg_ass_ord.md#function-pseudomatrix-algassvord-rngord)

    - [`PseudoMatrix(O): AlgAssVOrd[RngFunOrd]> → PMat`](alg_ass_ord.md#function-pseudomatrix-algassvord-rngfunord)

    - [`Basis(O): AlgAssVOrd → SeqEnum`](alg_ass_ord.md#function-basis-algassvord)

    - [`ZBasis(O): AlgAssVOrd[RngOrd] → [AlgAssVElt]`](alg_ass_ord.md#function-zbasis-algassvord-rngord)

    - [`ZBasis(O): AlgAssVOrd[RngFunOrd] → [AlgAssVElt]`](alg_ass_ord.md#function-zbasis-algassvord-rngfunord)

    - [`Generators(O): AlgAssVOrd → [AlgAssVElt]`](alg_ass_ord.md#function-generators-algassvord)

    - [`LocalBasis(O, p): AlgAssVOrd, RngOrdIdl → [AlgAssElt]`](alg_ass_ord.md#function-localbasis-algassvord-rngordidl)

    - [`Example: bases`](alg_ass_ord.md#example-ex-1fcab8)

  - [Predicates](alg_ass_ord.md#predicates)

    - [`O1 eq O2: AlgAssVOrd, AlgAssVOrd → BoolElt`](alg_ass_ord.md#operation-op-eq-algassvord-algassvord)

    - [`x in O: AlgAssVElt, AlgAssVOrd → BoolElt`](alg_ass_ord.md#operation-op-in-algassvelt-algassvord)

    - [`x notin O: AlgAssVElt, AlgAssVOrd → BoolElt`](alg_ass_ord.md#operation-op-notin-algassvelt-algassvord)

  - [Operations with Orders](alg_ass_ord.md#operations-with-orders)

    - [`Adjoin(O, x): AlgAssVOrd, AlgAssVElt → AlgAssVOrd`](alg_ass_ord.md#function-adjoin-algassvord-algassvelt)

    - [`Adjoin(O, x, I): AlgAssVOrd, AlgAssVElt, RngOrdFracIdl → AlgAssVOrd`](alg_ass_ord.md#function-adjoin-algassvord-algassvelt-rngordfracidl)

    - [`O1 + O2: AlgAssVOrd[RngOrd], AlgAssVOrd[RngOrd] → AlgAssVOrd`](alg_ass_ord.md#operation-op-plus-algassvord-rngord-algassvord-rngord)

    - [`O1 meet O2: AlgAssVOrd[RngOrd], AlgAssVOrd[RngOrd] → AlgAssVOrd`](alg_ass_ord.md#operation-op-meet-algassvord-rngord-algassvord-rngord)

    - [`O ^ x: AlgAssVOrd, AlgAssElt → AlgAssvOrd`](alg_ass_ord.md#operation-op-pow-algassvord-algasselt)

    - [`Example: sumandadjoin`](alg_ass_ord.md#example-ex-9c61c8)

- [Elements of Orders](elements.md)

  - [Construction of Elements](elements.md#construction-of-elements)

    - [`O ! 0: AlgAssVOrd, RngIntElt → AlgAssVOrdElt`](elements.md#operation-op-algassvord-rngintelt)

    - [`Zero(O): AlgAssVOrd → AlgAssVOrdElt`](elements.md#function-zero-algassvord)

    - [`O ! 1: AlgAssVOrd, RngIntElt → AlgAssVOrdElt`](elements.md#operation-op-algassvord-rngintelt-2)

    - [`One(O): AlgAssVOrd → AlgAssVOrdElt`](elements.md#function-one-algassvord)

    - [`O . i: AlgAssVOrd, RngIntElt → AlgAssVElt`](elements.md#operation-operation-algassvord-rngintelt-algassvelt)

    - [`O ! x: AlgAssVOrd, Any → AlgAssVOrdElt`](elements.md#operation-op-algassvord-any)

  - [Arithmetic of Elements](elements.md#arithmetic-of-elements)

    - [`x + y: AlgAssVOrdElt, AlgAssVOrdElt → AlgAssVOrdElt`](elements.md#operation-op-plus-algassvordelt-algassvordelt)

    - [`x - y: AlgAssVOrdElt, AlgAssVOrdElt → AlgAssVOrdElt`](elements.md#operation-op-minus-algassvordelt-algassvordelt)

    - [`- x: AlgAssVOrdElt → AlgAssVOrdElt`](elements.md#operation-operation-algassvordelt-algassvordelt)

    - [`x * y: AlgAssVOrdElt, AlgAssVOrdElt → AlgAssVOrdElt`](elements.md#operation-op-times-algassvordelt-algassvordelt)

    - [`u * c: AlgAssVOrdElt, RngElt → AlgAssVOrdElt`](elements.md#operation-op-times-algassvordelt-rngelt)

    - [`c * u: RngElt, AlgAssVOrdElt → AlgAssVOrdElt`](elements.md#operation-op-times-rngelt-algassvordelt)

    - [`x / y: AlgAssVOrdElt, AlgAssVOrdElt → AlgAssVElt`](elements.md#operation-op-div-algassvordelt-algassvordelt)

    - [`x div y: AlgAssVOrdElt, AlgAssVOrdElt → AlgAssVOrdElt`](elements.md#operation-operation-div-algassvordelt-algassvordelt-algassvordelt)

    - [`x ^ n: AlgAssVOrdElt, RngIntElt → AlgAssVOrdElt`](elements.md#operation-op-pow-algassvordelt-rngintelt)

  - [Predicates on Elements](elements.md#predicates-on-elements)

    - [`x eq y: AlgAssVOrdElt, AlgAssVOrdElt → BoolElt`](elements.md#operation-op-eq-algassvordelt-algassvordelt)

    - [`x ne y: AlgAssVOrdElt, AlgAssVOrdElt → BoolElt`](elements.md#operation-op-ne-algassvordelt-algassvordelt)

    - [`IsZero(x): AlgAssVOrdElt → BoolElt`](elements.md#function-iszero-algassvordelt)

    - [`IsUnit(a): AlgAssVOrdElt → BoolElt`](elements.md#function-isunit-algassvordelt)

    - [`IsScalar(x): AlgAssVOrdElt → BoolElt, RngElt`](elements.md#function-isscalar-algassvordelt)

  - [Other Operations with Elements](elements.md#other-operations-with-elements)

    - [`ElementToSequence(x): AlgAssVOrdElt → SeqEnum`](elements.md#function-elementtosequence-algassvordelt)

    - [`Eltseq(x): AlgAssVOrdElt → SeqEnum`](elements.md#function-eltseq-algassvordelt)

    - [`Norm(x): AlgAssVOrdElt → RngElt`](elements.md#function-norm-algassvordelt)

    - [`Trace(x): AlgAssVOrdElt → RngElt`](elements.md#function-trace-algassvordelt)

    - [`LeftRepresentationMatrix(e): AlgAssVOrdElt → AlgMatElt`](elements.md#function-leftrepresentationmatrix-algassvordelt)

    - [`RightRepresentationMatrix(e): AlgAssVOrdElt → AlgMatElt`](elements.md#function-rightrepresentationmatrix-algassvordelt)

    - [`RepresentationMatrix(a): AlgAssVOrdElt → AlgMatElt`](elements.md#function-representationmatrix-algassvordelt)

    - [`CharacteristicPolynomial(x): AlgAssVOrdElt → RngUPolElt`](elements.md#function-characteristicpolynomial-algassvordelt)

    - [`MinimalPolynomial(x): AlgAssVOrdElt → RngUPolElt`](elements.md#function-minimalpolynomial-algassvordelt)

- [Ideals of Orders](ideals.md)

  - [Construction of Ideals](ideals.md#construction-of-ideals)

    - [`lideal<O | E>: AlgAssVOrd, [AlgAssVOrdElt] → AlgAssVOrdIdl`](ideals.md#constructor-constructor-lideal-algassvord-algassvordelt-algassvordidl)

    - [`rideal<O | E>: AlgAssVOrd, [AlgAssVOrdElt] → AlgAssVOrdIdl`](ideals.md#constructor-constructor-rideal-algassvord-algassvordelt-algassvordidl)

    - [`ideal<O | E>: AlgAssVOrd, [AlgAssVOrdElt] → AlgAssVOrdIdl`](ideals.md#constructor-constructor-ideal-algassvord-algassvordelt-algassvordidl)

    - [`lideal<O | M>: AlgAssVOrd, PMat → AlgAssVOrdIdl`](ideals.md#constructor-constructor-lideal-algassvord-pmat-algassvordidl)

    - [`lideal<O | M>: AlgAssVOrd, Mtrx → AlgAssVOrdIdl`](ideals.md#constructor-constructor-lideal-algassvord-mtrx-algassvordidl)

    - [`rideal<O | M>: AlgAssVOrd, PMat → AlgAssVOrdIdl`](ideals.md#constructor-constructor-rideal-algassvord-pmat-algassvordidl)

    - [`rideal<O | M>: AlgAssVOrd, Mtrx → AlgAssVOrdIdl`](ideals.md#constructor-constructor-rideal-algassvord-mtrx-algassvordidl)

    - [`ideal<O | M>: AlgAssVOrd, PMat → AlgAssVOrdIdl`](ideals.md#constructor-constructor-ideal-algassvord-pmat-algassvordidl)

    - [`ideal<O | M>: AlgAssVOrd, Mtrx → AlgAssVOrdIdl`](ideals.md#constructor-constructor-ideal-algassvord-mtrx-algassvordidl)

    - [`O * e: AlgAssVOrd, RngElt → AlgAssVOrdIdl`](ideals.md#operation-op-times-algassvord-rngelt)

    - [`e * O: RngElt, AlgAssVOrd → AlgAssVOrdIdl`](ideals.md#operation-op-times-rngelt-algassvord)

  - [Attributes of Ideals](ideals.md#attributes-of-ideals)

    - [`Order(I): AlgAssVOrdIdl → AlgAssVOrd`](ideals.md#function-order-algassvordidl)

    - [`Algebra(I): AlgAssVOrdIdl → AlgAssV`](ideals.md#function-algebra-algassvordidl)

    - [`LeftOrder(I): AlgAssVOrdIdl → AlgAssVOrd`](ideals.md#function-leftorder-algassvordidl)

    - [`RightOrder(I): AlgAssVOrdIdl → AlgAssVOrd`](ideals.md#function-rightorder-algassvordidl)

    - [`ArithmeticRadical(O, p): AlgAssVOrd[RngOrd], RngOrdIdl → AlgAssVOrdIdl`](ideals.md#function-arithmeticradical-algassvord-rngord-rngordidl)

    - [`ArithmeticRadical(O, p): AlgQuat, RngElt → AlgQuatOrdIdl`](ideals.md#function-arithmeticradical-algquat-rngelt)

    - [`RadicalIdealizer(O, p): AlgAssVOrd[RngOrd], RngOrdIdl → AlgAssVOrd`](ideals.md#function-radicalidealizer-algassvord-rngord-rngordidl)

    - [`RadicalIdealizer(O, p): AlgQuat, RngElt → AlgQuat`](ideals.md#function-radicalidealizer-algquat-rngelt)

    - [`Neighbors(I, p): AlgAssVOrdIdl, RngOrdIdl → SeqEnum[AlgAssVOrdIdl]`](ideals.md#function-neighbors-algassvordidl-rngordidl)

    - [`Neighbors(I, p): AlgQuatOrdIdl, RngInt → SeqEnum[AlgQuatOrdIdl]`](ideals.md#function-neighbors-algquatordidl-rngint)

    - [`ElementsOfNorm(I, n): AlgAssVOrdIdl, RngElt → SeqEnum`](ideals.md#function-elementsofnorm-algassvordidl-rngelt)

  - [Bases of Ideals](ideals.md#bases-of-ideals)

    - [`Denominator(I): AlgAssVOrdIdl → RngElt`](ideals.md#function-denominator-algassvordidl)

    - [`PseudoBasis(I): AlgAssVOrdIdl[RngOrd] → SeqEnum`](ideals.md#function-pseudobasis-algassvordidl-rngord)

    - [`PseudoMatrix(I): AlgAssVOrdIdl[RngOrd] → PMat`](ideals.md#function-pseudomatrix-algassvordidl-rngord)

    - [`PseudoMatrix(I): AlgAssVOrdIdl[RngFunOrd] → PMat`](ideals.md#function-pseudomatrix-algassvordidl-rngfunord)

    - [`Basis(I): AlgAssVOrdIdl → SeqEnum`](ideals.md#function-basis-algassvordidl)

    - [`BasisMatrix(I): AlgAssVOrdIdl → AlgMatElt`](ideals.md#function-basismatrix-algassvordidl)

    - [`ZBasis(I): AlgAssVOrdIdl[RngOrd] → [AlgAssVOrdElt]`](ideals.md#function-zbasis-algassvordidl-rngord)

    - [`Generators(I): AlgAssVOrdIdl[RngOrd] → [AlgAssVOrdElt]`](ideals.md#function-generators-algassvordidl-rngord)

    - [`LocalBasis(I, p): AlgAssVOrdIdl, RngOrdIdl → [AlgAssElt]`](ideals.md#function-localbasis-algassvordidl-rngordidl)

    - [`PseudoBasis(I, R): AlgAssVOrdIdl[RngOrd], Str → SeqEnum`](ideals.md#function-pseudobasis-algassvordidl-rngord-str)

    - [`PseudoMatrix(I, R): AlgAssVOrdIdl[RngOrd], Str → PMat`](ideals.md#function-pseudomatrix-algassvordidl-rngord-str)

    - [`Basis(I, R): AlgAssVOrdIdl, Str → SeqEnum`](ideals.md#function-basis-algassvordidl-str)

    - [`BasisMatrix(I, R): AlgAssVOrdIdl, Str → AlgMatElt`](ideals.md#function-basismatrix-algassvordidl-str)

  - [Arithmetic for Ideals](ideals.md#arithmetic-for-ideals)

    - [`I + J: AlgAssVOrdIdl, AlgAssVOrdIdl → AlgAssVOrdIdl`](ideals.md#operation-op-plus-algassvordidl-algassvordidl)

    - [`I * J: AlgAssVOrdIdl, AlgAssVOrdIdl → AlgAssVOrdIdl, AlgAssVOrdIdl`](ideals.md#operation-op-times-algassvordidl-algassvordidl)

    - [`I * J: AlgAssVOrdIdl[RngOrd], AlgAssVOrdIdl[RngOrd] → AlgAssVOrdIdl, AlgAssVOrdIdl`](ideals.md#operation-op-times-algassvordidl-rngord-algassvordidl-rngord)

    - [`a * I: RngElt, AlgAssVOrdIdl → AlgAssVOrdIdl`](ideals.md#operation-op-times-rngelt-algassvordidl)

    - [`I * a: AlgAssVOrdIdl, RngElt → AlgAssVOrdIdl`](ideals.md#operation-op-times-algassvordidl-rngelt)

    - [`a * I: RngElt, AlgQuatOrdIdl → AlgQuatOrdIdl`](ideals.md#operation-op-times-rngelt-algquatordidl)

    - [`I * a: AlgQuatOrdIdl, RngElt → AlgQuatOrdIdl`](ideals.md#operation-op-times-algquatordidl-rngelt)

    - [`A * I: RngOrdFracIdl, AlgAssVOrdIdl[RngOrd]) → AlgAssVOrdIdl`](ideals.md#operation-op-times-rngordfracidl-algassvordidl-rngord)

    - [`I * A: RngOrdFracIdl, AlgAssVOrdIdl[RngOrd]) → AlgAssVOrdIdl`](ideals.md#operation-op-times-rngordfracidl-algassvordidl-rngord-2)

    - [`Colon(J, I): AlgAssVOrdIdl[RngOrd], AlgAssVOrdIdl[RngOrd] → PMat`](ideals.md#function-colon-algassvordidl-rngord-algassvordidl-rngord)

    - [`MultiplicatorRing(I): AlgAssVOrdIdl → AlgAssVOrd`](ideals.md#function-multiplicatorring-algassvordidl)

  - [Predicates on Ideals](ideals.md#predicates-on-ideals)

    - [`IsLeftIdeal(I): AlgAssVOrdIdl → BoolElt`](ideals.md#function-isleftideal-algassvordidl)

    - [`IsRightIdeal(I): AlgAssVOrdIdl → BoolElt`](ideals.md#function-isrightideal-algassvordidl)

    - [`IsTwoSidedIdeal(I): AlgAssVOrdIdl → BoolElt`](ideals.md#function-istwosidedideal-algassvordidl)

    - [`I eq J: AlgAssVOrdIdl, AlgAssVOrdIdl → BoolElt`](ideals.md#operation-op-eq-algassvordidl-algassvordidl)

    - [`I subset J: AlgAssVOrdIdl, AlgAssVOrdIdl → BoolElt`](ideals.md#operation-op-subset-algassvordidl-algassvordidl)

    - [`a in I: AlgAssVElt, AlgAssVOrdIdl → BoolElt`](ideals.md#operation-op-in-algassvelt-algassvordidl)

    - [`a notin I: AlgAssVElt, AlgAssVOrdIdl → BoolElt`](ideals.md#operation-op-notin-algassvelt-algassvordidl)

  - [Other Operations on Ideals](ideals.md#other-operations-on-ideals)

    - [`Norm(I): AlgAssVOrdIdl[RngOrd] → RngOrdIdl`](ideals.md#function-norm-algassvordidl-rngord)

    - [`Example: sumandadjoin`](ideals.md#example-ex-d618e7)