# ideal

© 2005 John Abbott
GNU Free Documentation License, Version 1.2

- Examples
- User documentation
  - Operations
    - Queries
  - Functions for ideals in polynomial rings
    - Monomial ideals
    - Using Frobby library
  - Writing new types of ideal
- Maintainer documentation for the classes ideal, IdealBase
- Bugs, Shortcomings and other ideas

CoCoALib Documentation Index

## Examples

- ex-RingHom3.C
- ex-AlexanderDual.C
- ex-QuotientBasis.C

## User documentation

The class `ideal` is for representing values which are ideals of some `ring`. There are several ways to create an `ideal`:

NOTE: THIS SYNTAX WILL PROBABLY CHANGE

- `ideal I(r)` -- `I` is the principal ideal generated by `r` (a `RingElem`) in the ring `owner(r)`
- `ideal I(r1, r2)` -- `RingElem`s in the same `ring`
- `ideal I(r1, r2, r3)` -- `RingElem`s in the same `ring`
- `ideal I(r1, r2, r3, r4)` -- `RingElem`s in the same `ring`
- `ideal I(R, gens)` -- `I` is the ideal of `R` generated by the `RingElem`s in the C++ vector<> `gens`, (all in the same `ring`)
- `ideal I(gens)` -- if `gens`=[] throws ERROR, otherwise equivalent to `I(owner(gens[0]), gens)`

  If you want to make an ideal in `R` with no generators use this syntax

         ideal(R, vector<RingElem>())

### Operations

The permitted operations on `ideal`s are: let `I` and `J` be two ideals of the same ring

- `I+J` -- the sum of two ideals
- `I += J` -- equivalent to `I = I+J`
- `intersection(I, J)` -- intersection of two ideals
- `colon(I, J)` -- the quotient of two ideals

  We may also enquire about certain properties of an ideal:
- `IsZero(I)` -- true iff the ideal is a zero ideal
- `IsOne(I)` -- true iff the ideal is the whole ring
- `IsMaximal(I)` -- true iff the ideal is maximal in its ring (i.e. iff the quotient ring is a field)
- `IsPrime(I)` -- true iff the ideal is prime (i.e. quotient ring has no zero-divisors)
- `IsContained(I, J)` -- true iff the ideal `I` is a subset of the ideal `J`
- `IsElem(r, I)` -- true iff `r` is an element of the ideal `I`
- `I == J` -- true iff the ideals are equal (their generating sets may be different)

- `AmbientRing(I)` -- the ring in which the ideal `I` resides
- `gens(I)` -- a C++ vector<> of `RingElem`s which generate `I`
- `TidyGens(I)` -- a C++ vector<> of `RingElem`s which generate `I` (this generating set is in some way "reduced", and will never contain a zero element)

#### Queries

It is also possible to give some information about an ideal:

       I->UserAssertsIsPrime()       to specify that ``I`` is known to be prime
       I->UserAssertsIsNotPrime()    to specify that ``I`` is known not to be prime
       I->UserAssertsIsMaximal()     to specify that ``I`` is known to be maximal
       I->UserAssertsIsNotMaximal()  to specify that ``I`` is known not to be maximal

Making an incorrect assertion using these functions may lead to a program crash, or at least to poorer run-time performance.

### Functions for ideals in polynomial rings

Additional functions for an ideal `I` in a `SparsePolyRing` `P`.

- `IsZeroDim(I)` -- true iff `I` is zero-dimensional
- `IsHomog(I)` -- true iff `I` is homogeneous
- `AreGensMonomial(I)` -- true iff given `gens(I)` are all monomial. NB `0` is NOT monomial
- `AreGensSquareFreeMonomial(I)` -- true iff given `gens(I)` are all monomial and radical. NB `0` is NOT monomial

- `GBasis(I)` -- same as `TidyGens` (stored into `I` for future use)
- `LT(I)` -- leading term ideal (also knows as initial ideal)
- `homog(h, I)` -- homogenization with the indeterminate `h`, a `RingElem`, indeterminate in `AmbientRing(I)`
- `QuotientBasis(I)` -- basis of the quotient as a `K`-vector space
- `PrimaryDecomposition(I)` -- only for square free monomial ideals (for now)

#### Monomial ideals

Additional functions for a monomial ideal `I` in a `SparsePolyRing`

- `PrimaryDecompositionMonId(I)` -- only for square free monomial ideals (for now)
- `AlexanderDual(I)` -- only for square free monomial ideals (for now)

#### Using Frobby library

- `PrimaryDecompositionFrobby(I)`
- `AlexanderDual(I)`, `AlexanderDual(I, pp)`
- and more...

### Writing new types of ideal

Anyone who writes a new type of ring class will have to consider writing a new type of ideal class to go with that ring. The ideal class must be derived from the abstract class IdealBase (and to be instantiable must offer implementations of all pure virtual functions). Be especially careful to update the data members IamPrime and IamMaximal in the non-const member functions (add, intersection, and colon).

Some guidance may be obtained from looking at the FieldIdealImpl class which implements ideals in a field (there are only two: ideal(0) and ideal(1)). See the file FieldIdeal.txt

## Maintainer documentation for the classes ideal, IdealBase

The class ideal is little more than a reference counting smart pointer class pointing to an object of type derived from IdealBase. This approach allows many different implementations of ideals to be manipulated in a convenient and transparent manner using a common abstract interface.

The abstract class IdealBase specifies the interface which every concrete ideal class must offer. It is more complicated than one might expect partly because we want to allow the advanced user to tell the ideal whether it has certain important properties (which might be computationally expensive to determine automatically).

## Bugs, Shortcomings and other ideas

The maintainer documentation is still quite incomplete.

Shouldn't ideals be created by a function called NewIdeal???

I am not at all sure about the wisdom of having implemented `IamPrime` and `IamMaximal`. It seems to be terribly easy to forget to update these values when ideal values are modified (e.g. in IdealBase::add). It has also led to rather more complication that I would have liked. BUT I don't see how to allow the user to state that an ideal is maximal/prime without incurring such complication.

Functions to examine the bool3 flags could be handy for heuristic short-cuts when an ideal is already known to have a certain property.

Is it worth having a constructor for principal ideals generated by a number rather than a RingElem? e.g. NewIdeal(R,5) or NewIdeal(R,BigInt(5)).

Several member functions have names not in accordance with the coding conventions.