# Ideals of Orders ## Construction of Ideals ### `lideal: AlgAssVOrd, [AlgAssVOrdElt] -> AlgAssVOrdIdl` ### `rideal: AlgAssVOrd, [AlgAssVOrdElt] -> AlgAssVOrdIdl` ### `ideal: AlgAssVOrd, [AlgAssVOrdElt] -> AlgAssVOrdIdl` For an associative order $O$, this constructs the left, right or two sided $O$-ideal generated by the elements in the given sequence $E$ (these elements should be coercible into $O$). ### `lideal: AlgAssVOrd, PMat -> AlgAssVOrdIdl` ### `lideal: AlgAssVOrd, Mtrx -> AlgAssVOrdIdl` ### `rideal: AlgAssVOrd, PMat -> AlgAssVOrdIdl` ### `rideal: AlgAssVOrd, Mtrx -> AlgAssVOrdIdl` ### `ideal: AlgAssVOrd, PMat -> AlgAssVOrdIdl` ### `ideal: AlgAssVOrd, Mtrx -> AlgAssVOrdIdl` Constructs a left, right or two sided ideal of the associative order $O$ whose basis is given by $M$, which may be either a matrix or a pseudo matrix. ### `O * e: AlgAssVOrd, RngElt -> AlgAssVOrdIdl` ### `e * O: RngElt, AlgAssVOrd -> AlgAssVOrdIdl` The principal left (right) ideal of the associative order $O$ generated by the element $e$. ## Attributes of Ideals In this section, $I$ is an ideal of some order in an associative algebra. ### `Order(I): AlgAssVOrdIdl -> AlgAssVOrd` For an ideal $I$ created as an ideal of an order $O$, this returns $O$. ### `Algebra(I): AlgAssVOrdIdl -> AlgAssV` This returns `Algebra(Order(I))`. ### `LeftOrder(I): AlgAssVOrdIdl -> AlgAssVOrd` ### `RightOrder(I): AlgAssVOrdIdl -> AlgAssVOrd` For an ideal $I$, this returns the (full) order in `Algebra(I)` which stabilizes $I$ under multiplication on the left or right respectively. Explicitly, the left order of $I$ is defined as $\{ x : x I \subseteq I \}$. ### `ArithmeticRadical(O, p): AlgAssVOrd[RngOrd], RngOrdIdl -> AlgAssVOrdIdl` ### `ArithmeticRadical(O, p): AlgQuat, RngElt -> AlgQuatOrdIdl` Given an order $O$ over a number ring $R$ and a prime ideal $p$ in $R$, return the *Arithmetic Radical* of $O$ over $p$. This is the unique two-sided ideal $J$ of $O$ over $pO$ such that $J/pO$ is the Jacobson radical of $O/pO$. ### `RadicalIdealizer(O, p): AlgAssVOrd[RngOrd], RngOrdIdl -> AlgAssVOrd` ### `RadicalIdealizer(O, p): AlgQuat, RngElt -> AlgQuat` ```magma Side: MonStgElt Default: "Intersection" ``` Returns either the left or right order of the arithmetical radical of $O$ at $p$ or the intersection of these two orders, depending on whether `Side` is `"Left"`, `"Right"` or `"Intersection"`. ### `Neighbors(I, p): AlgAssVOrdIdl, RngOrdIdl -> SeqEnum[AlgAssVOrdIdl]` ### `Neighbors(I, p): AlgQuatOrdIdl, RngInt -> SeqEnum[AlgQuatOrdIdl]` Returns the $p$-neighbors of the ideal $I$, namely the ideals $J \subseteq I$ such that ${\rm Nm}(J) = p {\rm Nm}(I)$. Currently, only implemented for squarefree ideals $p$ coprime to ${\rm Disc}(O) {\rm Nm}(I)$, where $O$ is `Order(I)`. ### `ElementsOfNorm(I, n): AlgAssVOrdIdl, RngElt -> SeqEnum` ```magma ModUnits: BoolElt Default: false ``` Returns the elements in $I$ of norm $n$. Currently, only implemented when $I$ is an ideal in a definite quaternion algebra. If `ModUnits` is set to `true`, returns only one representative from every orbit under the action of the group of units of norm $1$ in $O$, where $O$ is `Order(I)`. ## Bases of Ideals In this section, $I$ is an ideal of an order $O$ in an associative algebra. ### `Denominator(I): AlgAssVOrdIdl -> RngElt` This returns an element $d$ of `BaseRing(O)` such that $d I \subseteq O$, where $O$ is `Order(I)`. ### `PseudoBasis(I): AlgAssVOrdIdl[RngOrd] -> SeqEnum` Given an ideal $I$ of an order over $R$, where $R$ is an order in a number field, this returns data describing $I$ as an $R$-module. It returns a sequence of tuples $\langle I_i, b_i \rangle$ where $I_i$ is an ideal of $R$ and $b_i$ is an element of `Algebra(I)`, such that $I$ is the direct sum of the $R$-modules $I_i b_i$. In particular, the $b_i$ form a basis of `Algebra(I)` as a vector space over the field of fractions of $R$. Note that the $b_i$ may not lie in $I$. ### `PseudoMatrix(I): AlgAssVOrdIdl[RngOrd] -> PMat` ### `PseudoMatrix(I): AlgAssVOrdIdl[RngFunOrd] -> PMat` Returns the pseudomatrix corresponding to `PseudoBasis(I)`. ### `Basis(I): AlgAssVOrdIdl -> SeqEnum` When $I$ is an ideal of an order over ${\mathbb{Z}}$, this returns a ${\mathbb{Z}}$-module basis of $I$. When the base ring of `Order(I)` is an extension of ${\mathbb{Z}}$, this returns part of the data returned by `PseudoBasis(I)`: it returns a basis of `Algebra(I)`. Note that this determines nothing about $I$, while the full data returned by `PseudoBasis(I)` determines $I$. ### `BasisMatrix(I): AlgAssVOrdIdl -> AlgMatElt` This returns `Basis(I)` as a row matrix. ### `ZBasis(I): AlgAssVOrdIdl[RngOrd] -> [AlgAssVOrdElt]` Returns a ${\mathbb{Z}}$-module basis for the ideal $I$. ### `Generators(I): AlgAssVOrdIdl[RngOrd] -> [AlgAssVOrdElt]` Returns a sequence of generators for the ideal $I$ as a module over its base ring. ### `LocalBasis(I, p): AlgAssVOrdIdl, RngOrdIdl -> [AlgAssElt]` ```magma Type: MonStgElt Default: "" ``` Given an ideal $I$ of an $R$-order in an algebra $A$, this function returns a basis of a free $R$-module $F$ in $A$ such that $I$ and $F$ agree at the completion at the prime ideal $p$ of $R$. If `Type` is specified, it must either be `"Submodule"` or `"Supermodule"`. In which case $F$ is either a sub- or supermodule of $I$. ### `PseudoBasis(I, R): AlgAssVOrdIdl[RngOrd], Str -> SeqEnum` ### `PseudoMatrix(I, R): AlgAssVOrdIdl[RngOrd], Str -> PMat` ### `Basis(I, R): AlgAssVOrdIdl, Str -> SeqEnum` ### `BasisMatrix(I, R): AlgAssVOrdIdl, Str -> AlgMatElt` In these variants of the functions described above, the elements of the basis are expressed with respect to the basis of $R$, where $R$ is some ring contained in $A =$ `Algebra(I)` such that $R \otimes F = A$ where $F$ is the base field of $A$. ## Arithmetic for Ideals ### `I + J: AlgAssVOrdIdl, AlgAssVOrdIdl -> AlgAssVOrdIdl` The sum of the ideals $I$ and $J$, which are ideals which share a side in equal orders. ### `I * J: AlgAssVOrdIdl, AlgAssVOrdIdl -> AlgAssVOrdIdl, AlgAssVOrdIdl` ### `I * J: AlgAssVOrdIdl[RngOrd], AlgAssVOrdIdl[RngOrd] -> AlgAssVOrdIdl, AlgAssVOrdIdl` The product of the ideals $I$ and $J$, where $I$ is a right ideal and $J$ is a left ideal of the same order $O$. Returns the product given the structure of left and right ideal. ### `a * I: RngElt, AlgAssVOrdIdl -> AlgAssVOrdIdl` ### `I * a: AlgAssVOrdIdl, RngElt -> AlgAssVOrdIdl` ### `a * I: RngElt, AlgQuatOrdIdl -> AlgQuatOrdIdl` ### `I * a: AlgQuatOrdIdl, RngElt -> AlgQuatOrdIdl` Returns the product of $a$ and $I$ as an ideal. ### `A * I: RngOrdFracIdl, AlgAssVOrdIdl[RngOrd]) -> AlgAssVOrdIdl` ### `I * A: RngOrdFracIdl, AlgAssVOrdIdl[RngOrd]) -> AlgAssVOrdIdl` Returns the product of a fractional ideal $A$ of the base ring with $I$ as an ideal. ### `Colon(J, I): AlgAssVOrdIdl[RngOrd], AlgAssVOrdIdl[RngOrd] -> PMat` If $I,J$ are left ideals, returns the colon $(J:I)=\{x \in A: xI \subset J\}$, similarly defined if $I,J$ are right ideals. ### `MultiplicatorRing(I): AlgAssVOrdIdl -> AlgAssVOrd` Returns the colon $(I:I)$ of the ideal $I$, the set of all elements which multiply $I$ into $I$. ## Predicates on Ideals ### `IsLeftIdeal(I): AlgAssVOrdIdl -> BoolElt` ### `IsRightIdeal(I): AlgAssVOrdIdl -> BoolElt` ### `IsTwoSidedIdeal(I): AlgAssVOrdIdl -> BoolElt` Return `true` if the associative ideal $I$ is a left, right or two sided ideal (respectively). ### `I eq J: AlgAssVOrdIdl, AlgAssVOrdIdl -> BoolElt` Return `true` if the associative ideals $I$ and $J$ are equal. ### `I subset J: AlgAssVOrdIdl, AlgAssVOrdIdl -> BoolElt` Returns `true` if and only if the ideal $I$ is contained in the ideal $J$. ### `a in I: AlgAssVElt, AlgAssVOrdIdl -> BoolElt` ### `a notin I: AlgAssVElt, AlgAssVOrdIdl -> BoolElt` Return `true` (`false`) if the element $a$ of an associative algebra is contained in the associative ideal $I$. ## Other Operations on Ideals ### `Norm(I): AlgAssVOrdIdl[RngOrd] -> RngOrdIdl` Returns the norm of the ideal $I$, the ideal of the base number ring of $I$ generated by the norms of the elements in $I$. ### `Example: sumandadjoin (ex-d618e7)` ```magma > F := CyclotomicField(3); %%> // MaximalOrder is random > R := MaximalOrder(F); > A := Algebra(FPAlgebra); > O := Order([A.i : i in [1..9]]); > MinimalPolynomial(O.2); $.1^3 + 5/1*R.1 > I := rideal; > IsLeftIdeal(I), IsRightIdeal(I), IsTwoSidedIdeal(I); false true false > MultiplicatorRing(I) eq O; true > PseudoBasis(I); [ , , , , , , , , ] > ZBasis(I); [ [0 R.1 0 0 0 0 0 0 0], [0 R.2 0 0 0 0 0 0 0], [0 0 0 R.1 0 0 0 0 0], [0 0 0 R.2 0 0 0 0 0], [0 0 0 0 -R.1 - R.2 0 0 0 0], [0 0 0 0 R.1 0 0 0 0], [-5/1*R.1 0 0 0 0 0 0 0 0], [-5/1*R.2 0 0 0 0 0 0 0 0] ] > Norm(I); Principal Ideal of R Generator: 15625/1*R.1 > J := rideal; > Norm(J); Principal Ideal of R Generator: 729/1*R.1 > A!1 in I+J; false > Denominator(1/6*I); [1, 0] > Colon(J,I); Pseudo-matrix over Maximal Equation Order with defining polynomial x^2 + x + 1 over its ground order Principal Ideal of R Generator: 3/1*R.1 * ( R.1 0 0 0 0 0 0 0 0 ) Principal Ideal of R Generator: 3/1*R.1 * ( 0 R.1 0 0 0 0 0 0 0 ) Principal Ideal of R Generator: R.1 * ( 0 0 R.1 0 0 0 0 0 0 ) Fractional Principal Ideal of R Generator: 3/5*R.1 * ( 0 0 0 R.1 0 0 0 0 0 ) Principal Ideal of R Generator: R.1 * ( 0 0 0 0 R.1 0 0 0 0 ) Principal Ideal of R Generator: R.1 * ( 0 0 0 0 0 R.1 0 0 0 ) Fractional Principal Ideal of R Generator: -1/5*R.1 * ( 0 0 0 0 0 0 R.1 0 0 ) Principal Ideal of R Generator: R.1 * ( 0 0 0 0 0 0 0 R.1 0 ) Fractional Principal Ideal of R Generator: 1/5*R.1 * ( 0 0 0 0 0 0 0 0 R.1 ) ```