# QuotientRing

© 2005,2012 John Abbott, Anna M. Bigatti
GNU Free Documentation License, Version 1.2

CoCoALib Documentation Index

## User documentation for QuotientRing

A `QuotientRing` is an abstract class (inheriting from `ring`) representing quotients of a `ring` by an `ideal`.

See `RingElem` QuotientRing for operations on its elements.

### Examples

Extended example of use:

```    RingZZ ZZ = RingZZ();           // a copy of the ring of integers
ring ZZmod10 = NewZZmod(10);    // represents ZZ/(10) integers modulo 10
ring ZZmod10a = NewQuotientRing(ZZ, ideal(ZZ, 10)); // same as ZZmod10

RingHom phi = QuotientingHom(ZZmod10); // ring hom from ZZ to ZZmod10
RingElem r(ZZmod10, -3);               // an element of ZZmod10
RingElem preimage = CanonRepr(r);      // an element of ZZ = BaseRing(ZZmod10)

ring S = NewZZmod(2);              // another ring S, details do not matter much
RingHom theta = QuotientingHom(S); // any ring hom from ZZ to S will do instead
RingHom theta_bar = NewInducedHom(ZZmod10, theta); // induced ring hom from ZZmod10 to S

ring RmodI = NewQuotientRing(NewPolyRing(S,"x,y,z"), "x^2-2, y^3-1");
```

### Constructors and Pseudo-constructors

• `NewQuotientRing(R, I)` -- creates a new `ring` representing the quotient `R`/`I`. `I` must be an ideal of `R`; odd things may happen if `I`=`R`. If `I` is zero then the result is isomorphic to `R` but not equal to `R`; arithmetic in `R` is more efficient than arithmetic in `R`/ideal(0).
• `NewQuotientRing(R, str)` -- creates a new `ring` representing the quotient `R`/`I`, where `I` is the ideal in `R` generated by the list of `RingElem` in the string `str`.
• `NewZZmod(n)` -- creates a new `ring` representing the quotient `ZZ`/ideal(`n`) where `ZZ` is the ring of integers `RingZZ` and `n` is an integer. A CoCoALib error will be thrown if `n`=1 or `n`=-1. Currently an error will be thrown also if `n`=0 (see BUGS).
• `QuotientRing` -- sort of downcast the ring `R` to a quotient ring; will throw an `ErrorInfo` object with code `ERR::NotQuotRing` if needed.

#### NewZZmod or NewRingFp?

If `n` is a small prime then `NewZZmod(n)` produces the same result as `NewRingFp(n)` (or perhaps `NewRingFpDouble(n)`). If `n` is not a small prime then `NewRingFp(n)` throws an exception whereas `NewZZmod(n)` will produce a working quotient ring.

### Query and cast

• `IsQuotientRing(R)` returns true iff R is implemented as a `QuotientRing`
• `QuotientRingPtr(R)` returns a pointer to the quotient ring impl (for calling mem fns); will throw an `ErrorInfo` object with code `ERR::NotQuotRing` if needed.

### Operations on QuotientRing

In addition to the standard `ring` operations, a `QuotientRing` may be used in other functions.

Given `RmodI` a `QuotientRing` (representing R/I) built as `NewQuotientRing(R,I)` with `I` is an `ideal` of the `ring` `R`.

• `BaseRing(RmodI)` -- a reference to the base ring of `RmodI`, namely `R`
• `DefiningIdeal(RmodI)` -- a reference to the ideal `I` used to create `RmodI`
• `ReprRing(RmodI)` -- a reference to the ring used for internal representation of elements of `RmodI`

### Homomorphisms

• `QuotientingHom(RmodI)` -- a reference to the quotienting homomorphism from `R` to `RmodI`
• `InducedHom(RmodI, phi)` -- where `phi` is a `RingHom` from `R` to `S`, creates a new `RingHom` from `RmodI` to `S`. Two types of error may occur:
• `ERR::BadInducingHom` -- if `domain(phi)` is not `BaseRing(RmodI)`
• `ERR::BadInducingHomKer` -- if `phi` does not map the gens of `I` to zero.

## Maintainer documentation for QuotientRing, QuotientRingBase, GeneralQuotientRingImpl

While considering the design of this code it may help to keep in mind these two canonical implementations:

`GeneralQuotientRingImpl`
internally elements are represented as elements of a "representation ring" (which may differ from the base ring) which are kept reduced modulo some ideal (which may differ from the defining ideal)

`RingFpImpl`
internally elements are represented by machine integers (see doc for RingFpImpl), a representation incompatible with that used for elements of the ring of integers (which is probably the base ring)

`QuotientRingBase` is an abstract class derived from RingBase, and is the base class for all quotient rings. It adds the following four new pure virtual member functions which must be defined in every concrete quotient ring:

```      virtual RingElem myCanonRepr(ConstRawValue r) const;
virtual void myReduction(RawValue& image, ConstRawValue arg) const;
virtual const RingHom& myQuotientingHom() const;
virtual RingHom myInducedHomCtor(const RingHom& InducingHom) const;
```

The member function myCanonRepr has to return a copy of the value since we cannot be sure that the internal representation is compatible with the internal representation of elements of the base ring.

## Bugs, Shortcomings and other ideas

`IamTrueGCDDomain` always returns false. We can be clever in some easy cases, but it is hard in general (think of rings of algebraic integers which are gcd domains, but not euclidean domains).

Should NewZZmod(n) allow the case n==0? There's no mathematical reason to forbid it, but forbidding it may help detect programmer errors more quickly -- it seems unlikely that one would really want to quotient by ideal(0).

FAIRLY SERIOUS CONFUSION: the code seems to make REPEATED sanity checks see

```    QuotientRingBase::QuotientRingBase
NewQuotientRing

QuotientRingHomBase::QuotientRingHomBase
NewInducedHom
```

I suspect that the C++ ctors should use CoCoA_ASSERT instead of checking always (and throwing an exception).

FURTHER SERIOUS CONFUSION: there is ambiguity about the difference between myBaseRing and myReprRing esp. for creating induced homomorphisms: given ring R, and ring S = R/I, create ring T = S/J An induced hom from T should start from a hom with domain S; or is it reasonable to accept a hom with domain R? In this case for T myReprRing is R but myBaseRing is S.

Given a RingHom from a QuotientRing it is not generally possible to obtain a reference to an "inducing hom": consider the hom from ZZ/(2) to ZZ/(2)[x] created by CoeffEmbeddingHom. A RingHom equivalent to the inducing hom can be produced by composing QuotientingHom with the given hom.

20 March 2004: I have added the possibility of making a trivial ring by quotienting: previously this was disallowed for no good reason that I could discern.