
#include <ring.H>
Inheritance diagram for CoCoA::ring:
Public Member Functions  
ring (const RingBase *RingPtr)  
const RingBase *  operator> () const 
Allow const member fns to be called.  
const RingBase *  myRawPtr () const 
Used by "downcasting" functions IsRingFp, AsRingFp, etc. 
Copyright (c) 2005 John Abbott Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License, Version 1.2; with no Invariant Sections, no FrontCover Texts, and no BackCover Texts. A copy of the licence is included in the file COPYING in this directory. User documentation for the files ring.H and ring.C ================================================== The file ring.H introduces several classes used for representing rings and their elements. A normal user of the CoCoA library will use mostly the classes [ring] and [RingElem]. An object of type [ring] represents a mathematical ring with unity, and objects of type [RingElem] represent values from some ring. While the CoCoA library was conceived primarily for computing with commutative rings, the possibility of having certain noncommutative rings exists. A variable of type [ring] is effectively constant: its value may not be changed, and it cannot be assigned to. In contrast, a variable of type [RingElem] may change its value but only within the ring to which it is associated: this ring is specified when the variable is created. Here are a couple of code snippets to give an idea: VALID CODE INVALID CODE ring R = RingZ(); ring R; // no! RingElem a(R), b(R); a = 2; RingElem a = 2; // no! b = a+7; RingElem(2); // no! if (ab != 7) cout << "MISTAKE!\n"; There are fuller example programs in the "examples/" directory. Operations on Rings  The primary use for a variable of type [ring] is simply to specify the ring to which RingElem variables are associated. However, for some computations the best algorithm to use may depend on certain properties of the ring. Let R denote a variable of type [ring]: here are the properties you can ask about: characteristic(R)  returns the characteristic of R (as a ZZ) IsCommutative(R)  a boolean, true iff R is commutative IsIntegralDomain(R)  a boolean, true iff R has no zero divisors IsGCDDomain(R)  a boolean, true iff R is a GCD domain IsOrderedDomain(R)  a boolean, true iff R is arithmetically ordered IsField(R)  a boolean, true iff R is a field symbols(R)  a std::vector of the symbols in R (e.g. Q[x,y,z] contains the symbols x,y, and z) NOTE: a pragmatic approach is taken: e.g. [IsGCDDomain] is true only if GCDs can actually be computed using the CoCoA library. [IsQuotientRing] is true only if R is internally implemented as a quotient ring (e.g. Z/0Z is; Z is not). In general the function "IsXYZ" should be read as "Is internally implemented as XYZ". These query functions tell you how the ring is implemented: IsZ(R)  a boolean, true iff R is actually a ring of integers IsFractionField(R)  a boolean, true iff R is actually a field of fractions IsPolyRing(R)  a boolean, true iff R is actually a polynomial ring IsQuotientRing(R)  a boolean, true iff R is actually a quotient ring NOTE: Warning: for example if R = Z[x]/ideal(x) then R is obviously isomorphic to the ring of integers but IsZ(R) gives FALSE, and IsQuotientRing(R) gives TRUE. You can get the zero and one of a ring directly using the following: zero(R)  the zero element of R one(R)  the one element of R Operations on RingElems  A variable of type [RingElem] must be associated to a ring upon its creation. Thereafter the ring to which it is associated cannot be changed during the lifetime of the value, but the value inside the ring can be changed. [RingElems] are designed to be easy and safe to use; this incurs a certain runtime overhead, so a faster alternative is offered (see below in section "Fast and Ugly Code"). See also some example programs in the directory "examples/" for how to use [RingElems]. Constructors of [RingElem]s: assume R is a [ring], n a machine integer, and N a (big) integer of type [ZZ] RingElem r1(R);  an element of R, initially zero RingElem r2(R, n);  an element of R, initially the image of n RingElem r3(R, N);  an element of R, initially the image of N RingElem r4(r1);  a copy of r1, element of the same ring RingElem r4 = r1;  (alternative syntax, discouraged) zero(R)  the zero element of R one(R)  the one element of R Enquiries: let r1 be a (possibly const) [RingElem], and let N be a variable of type ZZ owner(r1)  the ring to which r1 is associated IsZero(r1)  true iff r1 is zero IsOne(r1)  true iff r1 is one IsMinusOne(r1)  true iff r1 is one IsInteger(N, r1)  true iff r1 is the image of an integer (if true, a preimage is placed in N) IsRational(N, D, r1)  true iff r1 is the image of a rational (if true, N/D becomes a preimage) IsUnit(r1)  true iff r1 has a multiplicative inverse IsDivisible(r1, r2)  true iff r1 is divisible by r2 raw(r1)  (see below, section "Fast and Ugly Code") Arithmetic: let r be a nonconst [RingElem], and r1, r2 be potentially const [RingElems]. Assume they are all associated to the same ring. Then the operations available are: (meanings are obvious) cout << r1  output value of r1 (decimal only, see notes) r1 == r2  equality test r1 != r2  notequal test r = r1  assignment r1  negation (unary minus) r1 + r2  sum r1  r2  difference r1 * r2  product r1 / r2  quotient, division must be exact (see IsDivisible) r += r1  equiv. r = r + r1 r = r1  equiv. r = r  r1 r *= r1  equiv. r = r * r1 r /= r1  equiv. r = r / r1 gcd(r1, r2)  an associate of the gcd (if ring is gcd domain) power(r1, n)  nth power of r1; n may be [long] or [ZZ] Ordering: if the ring is ordered then these functions may also be used. sign(r1)  value is 1, 0 or +1 according as r1 is negative, zero or positive [here r1 must belong to an ordered ring] abs(r1)  absolute value of r1 cmp(r1, r2)  returns a value <0, =0, >0 according as r1r2 is <0, =0, >0. r1 < r2  standard inequalities r1 > r2  ... r1 <= r2  ... r1 >= r2  ... Notes on operations  Operations on elements of different rings will cause a runtime error. In all functions involving two [RingElems] either r1 or r2 may be replaced by a machine integer (the type is actually [long]), or by a big integer (element of the class ZZ). The integer value is automatically mapped into the ring owning the [RingElem] in the same expression. The exponent n in the power function may be zero or negative, but a runtime error will be signalled if one attempts to compute a negative power of a noninvertible element or if one attempts to raise zero to the power zero. You cannot use [^] to compute powers  see Bugs section. An attempt to perform an inexact division or to compute a GCD not in a GCD domain will produce a runtime error. The printing of ring elements is always in decimal regardless of the [ostream] settings (this is supposed to be a feature rather than a bug). At this point, if you are new to CoCoALib, you should probably look at some of the example programs in the "examples/" directory. Writing functions with [RingElems] as arguments  One would normally expect to use the type [const RingElem&] for readonly arguments which are [RingElems], and [RingElem&] for readwrite arguments. Unfortunately, doing so would lead to problems with the CoCoA library. INSTEAD you should use the type [ConstRefRingElem] for readonly arguments, and the type [RefRingElem] for readwrite arguments. If you are curious to know why this nonstandard quirk has to be used, read on. Internally, ring element values are really smart pointers to the true value. Now the "const" keyword in C++ when applied to a pointer makes the pointer const while the pointedto value remains alterable  this is not the behaviour we want for [const RingElem&]. To get the desired behaviour we have to use another type: the type we haved called [ConstRefRingElem]. You might wonder why "Ref" appears in the names [RefRingElem] and [ConstRefRingElem]. It indicates that you are working with a reference to a value which is owned by another object (e.g. a variable of type [RingElem], or maybe a matrix). ADVANCED USE OF RINGS  The rest of this section is for more advanced use of rings and RingElems (e.g. by CoCoA library contributors). If you are new to CoCoA, you need not read beyond here. Writing C++ classes for new types of ring  A convention of the CoCoA library is that the class RingBase is to be used as an abstract base class for all rings. You are strongly urged to familiarize yourself with the maintainer documentation if you want to understand how and why rings are implemented as they are in the CoCoA library. Fast and Ugly Code  WE DO NOT RECOMMEND that you use what is described in this section. If you are curious to know a bit more how rings are implemented, you might find this section informative. RingElems are designed to be easy and pleasant to use, but this convenience has a price: a runtime performance penalty (and a memory space penalty too). Both penalities may be avoided by using "raw values" but at a considerable loss of programming convenience and safety. You should consider using raw values only if you are desperate for speed; even so, performance gains may be only marginal except perhaps for operations on elements of a simple ring (e.g. a small finite field). A RingElem object contains within itself an indication of the owning ring, and a "raw value" which is a pointer to where the real representation of the ring element value lies. These raw values may be accessed via the "raw" function. They may be combined arithmetically by calling member functions of the owning ring. For instance, if x,y,z are all RingElem objects all BELONGING TO EXACTLY THE SAME RING then we can achieve x = y+z; slightly faster by calling owner(x)>my.Add(raw(x), raw(y), raw(z)); It should now be clear that the syntax involved is cumbersome and somewhat obscure. For the future maintainability of the code the simpler "x = y+z;" has many advantages. Furthermore, should x,y,z somehow happen not all to lie in the same ring then "x = y+z;" will act in a reasonable way, whereas the supposedly faster call will likely lead to many hours of debugging grief. The member functions for arithmetic (e.g. myAdd) DO NOT PERFORM sanity checks on their arguments: e.g. attempting to divide by zero could well crash the program. If you use a "debugging version" of the CoCoA Library then some member functions do use assertions to check their arguments. This is useful during development, but must not be relied upon since the checks are absent from the "nondebugging version" of the CoCoA Library. See the file config.txt for more information. This fast, ugly, unsafe way of programming is made available for those who desperately need the speed. If you're not desperate, don't use it! Fast, Ugly and Unsafe operations on raw values  Read the section "Fast and Ugly Code" before using any of these! Let r be a nonconst raw value, and r1, r2 potentially const raw values. Assume they are all owned by the ring R. Then the functions available are: R>myNew()  "construct" a new element of R, value=0 R>myNew(n)  "construct" a new element of R, value=n R>myNew(N)  "construct" a new element of R, value=N R>myNew(r1)  "construct" a new element of R, value=r1 R>myDelete(r)  "destroy" r, element of R (frees resources) R>mySwap(r, s)  swaps the two values (s is nonconst raw value) R>myAssignZero(r)  r = 0 R>myAssign(r, r1)  r = r1 R>myAssign(r, n)  r = n (n is a long) R>myAssign(r, N)  r = n (N is a ZZ) R>myNegate(r, r1)  r = r1 R>myAdd(r, r1, r2)  r = r1+r2 R>mySub(r, r1, r2)  r = r1r2 R>myMul(r, r1, r2)  r = r1*r2 R>myDiv(r, r1, r2)  r = r1/r2 (division must be exact) R>myIsDivisible(r, r1, r2)  r = r1/r2, and returns true iff division was exact R>myIsUnit(r1)  IsUnit(r1) R>myGcd(r, r1, r2)  r = gcd(r1, r2) R>myPower(r, r1, n)  r = power(r1, n) BUT n MUST be nonnegative!! R>myIsZero(r1)  r1 == 0 R>myIsZeroAddMul(r, r1, r2)  ((r += r1*r2) == 0) R>myIsEqual(r1, r2)  r1 == r2 R>myOutput(out, r1)  out << r1 R>mySequentialPower(r, r1, n)  normally it is better to use R>myPower(r, r1, n) R>myBinaryPower(r, r1, n)  normally it is better to use R>myPower(r, r1, n) Maintainer documentation for the classes ring, RingBase and RingElem ==================================================================== (NB consider consulting also QuotientRing, FractionField and PolyRing) The design underlying rings and their elements is more complex than I would have liked, but it is not as complex as the source code may make it appear. The guiding principles are that the implementation should be flexible and easy/pleasant to use while offering a good degree of safety; extreme speed of execution was not a goal (as it is usually contrary to good flexibility) though an interface offering slightly better runtime efficiency remains. Regarding flexibility: in CoCoALib we want to handle polynomials whose coefficients reside in (almost) any commutative ring. Furthermore, the actual rings to be used will be decided at runtime, and cannot restricted to a given finite set. We have chosen to use C++ inheritance to achieve the implementation: the abstract class [RingBase] defines the interface that every concrete ring class must offer. Regarding ease of use: since C++ allows the common arithmetic operators to be overloaded, it is essential that these work as expected for elements of arbitrary rings  with the caveat that "/" means exact division, being the only reasonable interpretation. Due to problems of ambiguity arithmetic between elements of different rings is forbidden: e.g. let f in Q[x,y] and g in Z[y,x], where should f+g reside? The classes in the file ring.H are closely interrelated, and there is no obvious starting point for describing them  you may find that you need to read the following more than once to comprehend it. Here is a list of the classes: ring  value represents a ring; it is a smart pointer RingBase  abstract class "defining what a ring is" RingElem  value represents an element of a ring RefRingElem  reference to a RingElem ConstRefRingElem  constreference to a RingElem RingElemConstRawPtr  raw pointer to a "const" ring value RingElemRawPtr  raw pointer to a ring value The class [RingBase] is of interest primarily to those wanting to implement new types of ring (see relevant section below); otherwise you probably don't need to know about it. Note that [RingBase] includes an intrusive reference counter  so every concrete ring instance will have one. [RingBase] also includes a machine integer field containing a unique numerical ID  this is so that distinct copies of otherwise identical rings can be distinguished when output (e.g. in OpenMath). The class [ring] is used to represent mathematical rings (e.g. possible values include Z, Q, or Q[x,y,z]). An object of type [ring] is just a reference counting smart pointer to a concrete ring implementation object  so copying a ring is fairly cheap. In particular two rings are considered equal if and only if they refer to the same identical concrete ring implementation object. In other files you will find classes derived from [ring] which represent special subclasses of rings, for instance [PolyRing] is used to represent polynomial rings. The intrusive reference count, which must be present in every concrete ring implementation object, is defined as a data member of [RingBase]. The classes [RingElem], [RefRingElem] and [ConstRefRingElem] are related by inheritance: they are very similar but differ in important ways. The three classes are used for representing values in rings (e.g 1 as an element Z, or 1 as an element of Q[x], etc). In each case an object of that C++ type comprises two components: one is the identity of ring to which the element belongs, and the other is the value in that ring (the value is stored in a format that only the owning ring can comprehend). All operations on ring elements are effected by member functions of the ring to which the value belongs. The difference between [ConstRefRingElem] and [RefRingElem] is quite simple: you cannot change the value of a [ConstRefRingElem], for instance you cannot assign to it, while you are free to change the value of a [RefRingElem]. The difference between a [RefRingElem] and a [RingElem] is also quite simple, but possibly harder to grasp. A variable of type [RingElem] is the owner of the value that it represents: that value will be destroyed when the variable passes out of scope. In contrast, a variable of type [RefRingElem] is not the owner; it merely refers to value belonging to some other structure (e.g. a [RingElem], or a matrix, or a polynomial). So you can create a [RingElem] from nothing, whereas you must already have a ring element to be able to create a [RefRingElem] which refers to it. Why bother to distinguish between [RingElem] and [RefRingElem]? The main reason was to allow matrices and iterators of polynomials to be implemented cleanly and efficiently. Clearly a matrix should be the owner of the values appearing as its entries, but we also want a way of reading the matrix entries without having to copy them. Furthermore, the matrix can use a compact representation: the ring to which its elements belong is stored just once, and not once for each element. The reason that [ConstRefRingElem] and [RefRingElem] are distinct classes is that neither [const RefRingElem&] nor [const RingElem&] achieves what one might reasonably expect. Since a [RingElem] is effectively a pointer to the value represented, applying a C++ [const] keyword merely makes the pointer const while leaving the pointedto value modifiable. Consider the following procedure void CannotChange(const RefRingElem& x) { RefRingElem writable(x); // writable reference to value of x writable = writable+1; } The above procedure will add one to the value of its argument even though it would seem that it should not be alter the value. The inheritance structure between [ConstRefRingElem], [RefRingElem] and [RingElem] implements the similarities and differences between these classes while also allowing [ConstRefRingElem] and [RefRingElem] to be used as types of parameters to functions and procedures. Given that matrix entry accessors return a [ConstRefRingElem] it is important not to use [RingElem&] or [const RingElem&] as the parameter type because compilation would fail if a matrix entry were passed as parameter. As already hinted above, the internal data layouts for objects of types [RingElem], [RefRingElem] and [ConstRefRingElem] are identical  this is guaranteed by the C++ inheritance mechanism. The subfield indicating the ring to which the value belongs is simply a [const ring], which is just a reference counting smart pointer. The subfield indicating the value is a raw pointer of type [void* const]; however, when the raw pointer value is to be handled outside a ring element object then it is wrapped up as a [RingElemRawPtr] or [RingElemConstRawPtr]  these are simply wrapped copies of the [void*]. Make a careful note of the exact type of the data member [myValuePtr]: the pointer is constant while the pointed to value is not constant. The constness of the pointer is ABSOLUTELY CRUCIAL to the correct behaviour of [RefRingElem]. The fact that the pointedto value is not const may seem contradictory (for an object of type [ConstRefRingElem]), but it allows slightly easier implementation of the nonconstant derived classes [RefRingElem] and [RingElem]; the friend [raw] function puts in the necessary constness when it is called. The classes [RingElemRawPtr] and [RingElemConstRawPtr] are used for two reasons. One is that if a naked [void*] were used outside the ring element objects then C++ would find the call [RingElem(R, 0)] ambiguous because the constant [0] can be interpreted either as an integer constant or as a null pointer: there are two constructors which match the call equally well. The other reason is that it discourages accidentally creating a ring element object from any old pointer; it makes the programmer think  plus I feel uneasy when there are naked [void*] pointers around. Note that the type of the data member [myPtr] is simply [void*] as opposed to [void const*] which one might reasonably expect. I implemented it this way as it is simpler to add in the missing constness in the member function [RingElemConstRawPtr::myRawPtr] than it would be to cast it away in the [myRawPtr] function of [RingElemRawPtr]. In [ConstRefRingElem] why did I chose to make the data member [myValuePtr] of type [void* const] rather than [RingElemRawPtr const]? Further comments about implementation aspects of the above classes. Recall that [ring] is essentially a smart pointer to a [RingBase] object, i.e. concrete implementation of a ring. Access to the implementation is given via [operator>]. If necessary, the pointer may also be read using the member function [myRingPtr]: this is helpful for defining functions such as [IsPolyRing] where access to the pointer is required but [operator>] cannot be used. The class [RingBase] declares a number of pure virtual functions for computing with ring elements. Since these functions are pure they must all be fully defined in any instantiable ring class (e.g. RingZImpl or RingFpImpl). These member functions follow certain conventions: * RETURN VALUES: most arithmetic functions return no value, instead the result is placed in one of the arguments (normally the first argument is the one in which the result is placed), but functions which return particularly simple values (e.g. booleans or machine integers) do indeed return the values by the usual function return mechanism. * ARG TYPES: ring element values are passed as "raw pointers" (i.e. a wrapped [void*] pointing to the actual value). A readonly arg is of type [RingElemConstRawPtr], while a writable arg is of type [RingElemRawPtr]. When there are writable args they normally appear first. For brevity there are typedefs [ConstRawPtr] and [RawPtr] in the scope of [RingBase] or any derived class. * ARG CHECKS: sanity checks on the arguments are NOT CONDUCTED (e.g. the division function assumes the divisor is nonzero). These member functions are supposed to be fast rather than safe. In a few cases there are nonpure virtual member functions in [RingBase]. They exist either because there is a simple universal definition or merely to avoid having to define inappropriate member functions (e.g. gcd functions when the ring cannot be a gcd domain). Here is a list of them: IamGCDDomain()  defaults to true IamOrderedDomain  defaults to false myIsUnit(x)  by default checks that 1 is divisible by x myGcd(lhs, x, y)  gives an error: either NotGcdDom or NYI myGcdQuot(lhs, xquot, yquot, x, y)  gives an error: either NotGcdDom or NYI myExgcd(lhs, xcofac, ycofac, x, y)  gives an error: either NotGcdDom or NYI myIsPrintAtom(x)  defaults to false myIsPrintedWithMinus(x)  gives a SERIOUS error myIsMinusOne(x)  defaults to myIsOne(x); calculates x myIsZeroAddMul(lhs, y, z)  computes lhs += y*z in the obvious way, and calls myIsZero myCmp(x, y)  gives NotOrdDom error mySign(x)  simply calls myCmp(x, 0), then returns 1,0,1 accordingly There are three nonvirtual member functions for calculating powers: one uses the sequential method, the other two implement the repeated squaring method (one is an entry point, the other an implementation detail). These are nonvirtual since they do not need to be redefined; they are universal for all rings. For the moment I shall assume that the intended meaning of the pure virtual functions is obvious (given the comments in the source code). Recall that arithmetic operations on objects of type [RingElem], [RefRingElem] and [ConstRefRingElem] are converted into member function calls of the corresponding owning ring. Here is the source code for addition of ring elements  it typifies the implementation of operations on ring elements. RingElem operator+(ConstRefRingElem x, ConstRefRingElem y) { const ring& Rx = owner(x); const ring& Ry = owner(y); if (Rx != Ry) error(CoCoAError(ERR::MixedRings, "RingElem + RingElem")); RingElem ans(Rx); Rx>myAdd(raw(ans), raw(x), raw(y)); return ans; } The arguments are of type [ConstRefRingElem] since they are readonly, and the return type is [RingElem] since it is new selfowning value (it does not refer to a value belinging to some other structure]. Inside the function we check that the rings of the arguments are compatible, and report an error if they are not. Otherwise a temporary local variable is created for the answer, and the actual computation is effected via a member function call to the ring in which the values lie. Note the use of the [raw] function for accessing the raw pointer of a ring element. In summary, an operation on ring elements intended for public use should fully check its arguments for compatibility and correctness (e.g. to avoid division by zero); if all checks pass, the result is computed by passing raw pointers to the appropriate member functions of the ring involved  this member function assumes that the values handed to it are compatible and valid; if not, "undefined behaviour" will result (i.e. a crash if you are lucky). Most of the member functions of a ring are for manipulating raw values from that same ring, a few permit one to query properties of the ring. The type of a raw value is RingBase::RawValue, which helpfully abbreviates to RawValue inside the namespace of RingBase. Wherever possible the concrete implementations should be "exception safe", i.e. they should offer either the strong exception guarantee or the nothrow guarantee (according to the definitions in "Exceptional C++" by Sutter). Bugs, Shortcomings and other ideas ================================== Shouldn't there be hints and notes about creating your own new type of concrete ring??? I have chosen not to use [operator^] for computing powers because of a significant risk of misunderstanding between programmer and compiler. The syntax/grammar of C++ cannot be changed, and [operator^] binds less tightly than (binary) [operator*], so any expression of the form [a*b^c] will be parsed as [(a*b)^c]; this is almost certainly not what the programmer intended. To avoid such problems of misunderstanding I have preferred not to define [operator^]; it seems too dangerous. How to swap [RefRingElem]s? Must be careful when trying to swap a [RefRingElem] with a [RingElem] to avoid possible orphans (memory leaks) or doubly owned values. Note about comparison operators (<,<=,>,>=, and !=). The C++ STL does have templates which will define all the relational operators efficiently assuming the existence of [operator<] and [operator==]. These are defined in the namespace [std::rel_ops] in the standard header file <utility>. I have chosen NOT to use these because they can define only "homogeneous" comparisons; so the comparisons between [ConstRefRingElem] and [int] or [ZZ] would still have to be written out manually, and I prefer the symmetry of writing them all out. See p.69ff of Josuttis for details. Printing rings is unsatisfactory. Need a mechanism for specifying a print name for a ring; and also a mechanism for printing out the full definition of the ring avoiding all/some print names. For instance, R = Q(x), S = R[a,b]; S could print as S, R[a,b] or Q(x)[a,b]. We should allow at least the first and the last of these possibilities. The function [myAssignZero] was NECESSARY because [myAssign(x, 0)] was ambiguous (ambiguated by the assignment from an [mpz_t]). It is no longer necessary, but I prefer to keep it (for the time being). The requirement to use the type [ConstRefRingElem] for function arguments (which should normally be [const RingElem&] is not ideal, but it seems hard to find a better way. It is not nice to expect users to use a funny type for their function arguments. How else could I implement access to coefficients in a polynomial via an iterator, or access to matrix elements? Would we want ++ and  operators for RingElems??? I had hoped to place the auto_ptrs [myZeroPtr] and [myOnePtr] in RingBase, but this caused trouble when rings are destroyed: the concrete ring was destroyed before the zero and one elements were destroyed. It seems much safer simply to duplicate the code for each ring implementation class. Should (some of) the query functions return [bool3] values? What about properties which are hard to determine? There used to be an [IsFinite(R)] function in the documentation, but it did not exist in the code. Should it exist? How to generate random elements from a ring? Note the slightly unusual return types for [operator+=] etc; it seemed daft to have a referencetoreference. In contrast, I left the return type for [RingElem::operator=] as [RingElem&]; it felt very odd thinking of some other type [RefRingElem] as the return type. The dtor for [ConstRefRingElem] is deliberately not virtual; idem for [RefRingElem]. This could potentially cause trouble if you convert a pointer to [RingElem] into a pointer to [RefRingElem]; but if you do that, you'd better be doubly sure about what you're doing anyway.  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 Q[gensym], or DUP extension Q[gensym] Q[gensym] > quotient by gensym^22 to get Q(sqrt(2)) Q(sqrt(2)) > polynomial ring (3 vars) 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^22) to be regarded as isomorphically equivalent to Q[y]/(y^22)? 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^22), and S = Q[x]/(3618x^2), and T = Q[x]/(x^22). 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) = FF(Z[alpha]) though denoms in Q(alpha) can be taken in Z Q[alpha]/I_alpha = FF(Z[alpha]/I_alpha) **BUT** the ideal on LHS is an ideal inside Q[alpha] whereas that in RHS is in Z[alpha]. Furthermore Z[alpha]/I_alpha is "hairy" if the min poly of alpha is not monic!
Definition at line 54 of file ring.H.



Allow const member fns to be called.
Reimplemented in CoCoA::FractionField, CoCoA::PolyRing, CoCoA::QuotientRing, CoCoA::RingFloat, and CoCoA::SparsePolyRing. Definition at line 60 of file ring.H. Referenced by CoCoA::SparsePolyRing::operator>(), CoCoA::RingFloat::operator>(), CoCoA::QuotientRing::operator>(), CoCoA::PolyRing::operator>(), and CoCoA::FractionField::operator>(). 

Used by "downcasting" functions IsRingFp, AsRingFp, etc.
Definition at line 61 of file ring.H. References CoCoA::SmartPtrIRC< T >::myRawPtr(). Referenced by CoCoA::AsPolyRing(), CoCoA::AsSparsePolyRing(), CoCoA::IsPolyRing(), CoCoA::IsSparsePolyRing(), and CoCoA::operator==(). 