# RingHom

© 2005,2010 John Abbott
GNU Free Documentation License, Version 1.2

- User documentation for the files RingHom.H and RingHom.C
  - Examples
  - Constructors
    - Embeddings
    - Induced homomorphisms
    - Homomorphisms on polynomial rings
  - Applying a RingHom
  - Composition
  - Domain and Codomain
  - Kernel
  - Member Functions for Operations on Raw Values
- Maintainer documentation for the files RingHom.H and RingHom.C
- Bugs, Shortcomings and other ideas
- Some very old notes about implementing rings
  - Mapping elements between rings automatically

CoCoALib Documentation Index

## User documentation for the files RingHom.H and RingHom.C

The class `RingHom` is intended to represent homomorphisms between rings. Currently there is no way to represent more general maps between rings. It is possible to create a partial homomorphism which can generate run-time errors when applied to certain values.

The main operations available for `RingHom`s are application to a `RingElem`, and composition: both operations use function application syntax (see below for details).

### Examples

- ex-RingHom1.C
- ex-RingHom2.C
- ex-RingHom3.C
- ex-RingHom4.C
- ex-RingHom5.C

### Constructors

Here is a complete list of pseudo-constructors for ring homomorphisms (some are defined in other files, e.g. `QuotientRing.H` or `FractionField.H`). You should check also the documentation for `CanonicalHom` which tries to offer an easy way of building certain simple canonical homomorphisms.

- `IdentityHom(R)` -- where `R` may be any `ring`, gives the identity homomorphism on `R`

#### Embeddings

- `ZZEmbeddingHom(R)` -- gives the unique homomorphism from `RingZZ` to the `ring` `R`

- `QQEmbeddingHom(R)` -- may be partial gives the unique homomorphism from `RingQQ` to the `ring` `R`. Note that the resulting homomorphism may be only partial (e.g. if Characteristic(R) is not 0, or if the codomain is not a field).

- `CoeffEmbeddingHom(P)` -- where `P` is a `PolyRing`, gives the embedding homomorphism from the coefficient ring into the polynomial ring.

- `EmbeddingHom(FrF)` -- where `FrF` is a `FractionField`, gives the embedding homomorphism from the base ring into the fracion field (i.e. x |-> x/1)

#### Induced homomorphisms

- `InducedHom(RmodI, phi)` -- where `RmodI` is a `QuotientRing`, gives the homomorphism R/I --> S induced by `phi`: R --> S (which must have the base ring of `RmodI` as its domain, and whose kernel must contain the defining ideal of `RmodI`)

- `InducedHom(FrF, phi)` -- may be partial where `FrF` is a `FractionField`, gives the homomorphism induced by `phi` (which must have the base ring of `FrF` as its domain). Note that the resulting homomorphism may be only partial (e.g. if ker(`phi`) is non-trivial, or if the codomain is not a field).

#### Homomorphisms on polynomial rings

- `PolyAlgebraHom(Rx, Ry, xImages)` -- where
  - `Rx` and `Ry` are `PolyRing` with the same CoeffRing
  - `xImages` is a `vector` of `RingElem`
  gives the homomorphism from `Rx` to `Ry` mapping coefficients into themselves and mapping the k-th indeterminate of `Rx` to the k-th value in `xImages` (i.e. having index k-1)

- `PolyRingHom(Rx, S, CoeffHom, xImages)` -- where
  - `Rx` is a `PolyRing`
  - `CoeffHom` is a homomorphism whose domain is `CoeffRing(Rx)` and codomain is `S` or CoeffRing(S) (if `S` is a `PolyRing`)
  - `xImages` is a `vector` of `RingElem`
  gives the homomorphism from `Rx` to `S` mapping coefficients according to `CoeffHom` and mapping the k-th indeterminate of `Rx` to the k-th value in `xImages` (i.e. having index k-1)

### Applying a RingHom

A `RingHom` may applied using natural syntax: - let phi be an object of type `RingHom` - let x be an object of type `RingElem` - let n be of type `long` or `int` - let N be an object of type `BigInt` - let q be an object of type `BigRat`

        phi(x)  applies phi to x; error if owner(x) != domain(phi)
        phi(n)  applies phi to the image of n in domain(phi)
        phi(N)  applies phi to the image of N in domain(phi)
        phi(q)  applies phi to the image of q in domain(phi)

In all cases the result is a `RingElem` belonging to the codomain of phi. Currently partial homomorphisms are allowed, so applying a RingHom could trigger an error (e.g. an induced hom from Q to Z/(3) applied to 1/3).

If the `RingElem` supplied belongs to the wrong `ring` (i.e. not the domain of the `RingHom`) then an exception is thrown with code `ERR::BadRingHomArg1`. If the argument belongs to the right `ring` but lies outside the domain then an exception with code `ERR::BadRingHomArg2` is thrown.

### Composition

Two `RingHom`s may be composed using a fairly natural syntax: if we have two `RingHom`s phi:R -> S and theta:S -> T then their composition may be computed using the syntax

       theta(phi)   the composite homomorphism "apply phi first then theta"

If the domain of the outer homomorphism is different from the codomain of the inner homomorphism then an exception is thrown with code `ERR::BadCompose`.

### Domain and Codomain

We may ask for the domain and codomain of a `RingHom` phi:

        domain(phi)       gives a const ref to the domain
        codomain(phi)     gives a const ref to the codomain

Note that the domain and codomain are merely `ring`s, they "forget" any special ring type (such as `SparsePolyRing`).

### Kernel

Currently it is not possible to ask for the kernel of a `RingHom`.

### Member Functions for Operations on Raw Values

All operations on a `RingHom` are invisibly converted into member function calls on a `RingHomBase`. It is possible to call these member functions directly: the main difference is that the member functions do not perform any sanity checking on their arguments (so they should be slightly faster but if you hand in incompatible arguments, you'll probably get an ugly crash).

## Maintainer documentation for the files RingHom.H and RingHom.C

These files contain two "generic" classes (`RingHom` and `RingHomBase`), and a trivial concrete class representing the identity ring homomorphism, `IdentityRingHom`. Most of this section is dedicated to the two generic classes since they represent the primary contribution to the CoCoA library.

The class `RingHom` is little more than a "reference counting smart pointer" class to objects of type `RingHomBase`; this latter type is designed to support intrusive reference counting. Beyond its role as a smart pointer `RingHom` offers four "function application" syntaxes:

        RingElem RingHom::operator()(ConstRefRingElem x) const;
        RingElem RingHom::operator()(long n) const;
        RingElem RingHom::operator()(const BigInt& N) const;
        RingHom RingHom::operator()(const RingHom&) const;

The first three support a natural syntax for applying the homomorphism to a ring element, a small integer, or a large integer. The last offers a fairly natural syntax for creating the composition of two homomorphisms.

The class `RingHomBase` is a purely abstract class which is used to specify the interface which any concrete ring homomorphism class must offer. In particular this base class already includes an intrusive reference counter, as required by `RingHom`. It also includes two private data members `myDomainValue` and `myCodomainValue` which store the domain and codomain rings. Note that these data fields are plain `ring`s and so "forget" any special ring type which the domain or codomain may have had. Originally I had hoped to preserve any special ring type information, but this seemed to lead to a confusing and complex implementation (which probably would never have worked as I hoped). The two `ring` fields may be read using the accessor functions:

          const ring& myDomain() const;
          const ring& myCodomain() const;

A concrete class implementing a ring homomorphism must supply definition for the following (pure virtual) functions:

          virtual void myApply(RingBase::RawValue& image, RingBase::ConstRawValue arg) const;
          virtual void myOutputSelf(std::ostream& out) const;

DO NOTE THAT the two arguments to `myApply` normally belong to DIFFERENT rings. `arg` belongs to `myDomain()` whereas `image` belongs to `myCodomain()`. The function `myOutputSelf` should print out a useful description of the homomorphism.

## Bugs, Shortcomings and other ideas

Cannot compute a kernel of a RingHom.

Arranging for `domain(phi)` and `codomain(phi)` to preserve C++ type information about the respective rings (e.g. `PolyRing` or `FractionField` rather than simply `ring`), appears to be difficult to achieve in any reasonable manner. I've decided that it is much simpler just to discard all special type information, and return simply `ring`s. If the user knows something more, he can use a "cast" function like `AsFractionField`. Even if it were feasible to maintain such C++ type info, there would have to n-squared cases to cover all possible combinations of domain and codomain.

We should implement more special cases: e.g. same vars different coeff ring, PP --> PP, other... Also need some way of handling canonical homomorphisms.

Some special cases of homomorphic embeddings R --> S: (may belong with the special types of ring to which they are associated)

- (a) S is an identical copy of R
- (b) S is the same (poly)ring as R but with a different term ordering
- (c) R, S are the same polynomial ring (same vars and ordering) but with different coefficients
- (d) each generator of R maps to a power product (or 0) in S
- (e) S is the same as R but with more variables (is also of type (d))
- (f) permutation of the variables (is also of type (d))
- (g) general homomorphism mapping
- (h) S is the fraction field of R

## Some very old notes about implementing rings

This all needs to be sorted out!

### Mapping elements between rings automatically

How to decide whether a value can be mapped into the current_ring?

If the rings are marked as being equivalent isomorphically then we can just use the obvious isomorphism. A more interesting case is when a value resides in a ring which is a natural subring of the current_ring e.g. Z inside Q(sqrt(2))[x,y,z].

One could argue that to create `Q(sqrt(2))[x,y,z]` we had to follow this path

- `Z` --> fraction field `Q`
- `Q` --> polynomial ring (1 indet) or DUP extension `Q[gensym]`
- `Q[gensym]` --> quotient by gensym^2-2 to get `Q(sqrt(2))`
- `Q(sqrt(2))` --> polynomial ring (3 indets) `Q(sqrt(2))[x,y,z]`

From this it ought to be easy to identify natural embeddings of `Z`, `Q`, and (possibly) `Q(sqrt(2))` in `Q(sqrt(2))[x,y,z]`. We do not get an embedding for Q[gensym] since we had to generate the symbol gensym and no one else can create the same gensym. Because of this it is not altogether clear that an independently created copy of `Q(sqrt(2))` can be embedded automatically, since that copy would have a different symbol/gensym. Now if the algebraic extension were achieved directly...

Would we want `Q[x]/(x^2-2)` to be regarded as isomorphically equivalent to `Q[y]/(y^2-2)`? In fact there are two possible isoms: `x <---> y` and `x <---> -y`. I think that these should not be viewed as isom automatically, especially as there is more than one possible choice.

In contrast, if `R = Q[x]/(x^2-2)`, and `S = Q[x]/(36-18x^2)`, and `T = Q[x]/(x^2-2)`. It is clear that `Q[x]` can be mapped into each of `R`, `S` and `T` in a natural way. Of course, in each case `x` stands for sqrt(2), and it wouldn't be too hard to spot that `R` and `T` are identical; it is not quite as simple to see that `R` and `S` are isom. Presumably with a little more effort one could create examples where it could be jolly hard to spot that two such rings are just the same ring. For this reason, I think no attempt should be made to spot such natural isoms between independent rings. Had `T` been created from `R` (e.g. by making copy via assignment) then they would no longer be independent, and a natural isom could be deduced automatically. Now I think about it, a facility to make a copy of a ring WITHOUT the natural isom should be made available.

There is also a need for a way to specify that one ring embeds naturally into another (and via which homomorphism), or indeed that they are isomorphic. Isomorphism could be expressed by giving two inverse homs -- the system could then check that the homs are inverse on the generators, how it would check that the maps are homs is not so clear (perhaps the only maps which can be created are homs). Oooops, this would allow one to declare that `Z` and `Q` (or `Z[x]` and `Q[x]`) are isom..... need to think more about this!

A similar mechanism will be needed for modules (and vector spaces). A module should naturally embed into a vector space over the fraction field of the base ring....

Conceivably someone might want to change the natural embedding between two rings. So a means of finding out what the natural embedding is will be necessary, and also a way replacing it.

There is also a general question of retracting values into subrings. Suppose I have computed 2 in `Q(x)`, can I get the integer 2 from this? In this case I think the user must indicate explicitly that a retraction is to occur. Obviously retraction can only be into rings on the way to where the value currently resides.

Other points to note:

      Q(x) = Z(x) = FrF(Z[x]) == FrF(FrF(Z)[x])

Q(alpha) = FrF(Z[alpha]) though denoms in Q(alpha) can be taken in Z

Q[alpha]/I_alpha = FrF(Z[alpha]/I_alpha) BUT the ideal on LHS is an ideal inside Q[alpha] whereas that on RHS is in Z[alpha]. Furthermore Z[alpha]/I_alpha is hairy if the min poly of alpha is not monic!