© 2005-2010,2012 John Abbott, Anna M. Bigatti
GNU Free Documentation License, Version 1.2

CoCoALib Documentation Index


User documentation

The file ring.H introduces several classes used for representing rings and their elements. A normal user of the CoCoA library will use principally 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. To make the documentation more manageable it has been split into two: this file describes operations on a RingElem, whereas a separate file describes the operations directly applicable to rings. Documentation about the creation and use of homomorphisms between rings can be found in RingHom.

An object of type RingElem comprises two internal parts: the ring to which the value belongs, and the value itself. For instance, this means that the zero elements of different rings are quite different objects.


Normally when creating a new RingElem we specify both the ring to which it belongs, and its initial value in that ring. Let R be a ring. Let n be a machine integer or a BigInt. Let q be a rational, i.e. a value of type BigRat. Let r2 be a ring element.

RingElem r; the zero element of RingZZ() (special case)
RingElem r(R); an element of R, initially 0
RingElem r(R, n); an element of R, initially the image of n
RingElem r(R, q); an element of R, initially the image of q (or error)
RingElem r(R, s); an element of R, initially the value of symbol s
RingElem r(R, r2); an element of R, maps r2 into R via CanonicalHom
RingElem r(R, MPZ); an element of R, initially the value of mpz_t MPZ
RingElem r(R, MPQ); an element of R, initially the value of mpq_t MPQ (or error)

Note 1: To create a RingElem from a value of type mpz_class you must do RingElem(R, MPZ.get_mpz_t()); analogously for mpq_class.

Note 2: Construction from a rational may fail, e.g. if the denominator is a zero divisor in the ring; if it does fail then an exception is thrown (with code ERR::DivByZero).

You can create a copy of a ring element in the usual way:

RingElem r(r2); a copy of r2, element of the same ring
RingElem r = r2; (alternative syntax, discouraged)

Naturally the last constructor works only if the denominator of the rational q is 1.

These are not really constructors: 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

RingElems are designed to be easy and safe to use; the various checks do incur a certain run-time overhead, so a faster alternative is offered (see below in the section Fast and Ugly Code). Arithmetic operations between RingElems will fail if they do not belong to the same ring (the exception has code ERR::MixedRings).

Assignment & Swapping

Assigning an integer or rational to a RingElem wil automatically map the value into the ring to which the RingElem belongs.

r = n; map n into owner(r) and assign the result
r = q; map q into owner(r) and assign the result
r = r2; r becomes a copy of r2 -- afterwards owner(r)==owner(r2)
swap(r,s); exchange the values (and the owning rings)


Arithmetic operations between RingElems will fail if they do not belong to the same ring (the exception has code ERR::MixedRings). You may perform arithmetic between a RingElem and a machine integer, a BigInt value or a BigRat value -- the integer/rational is automatically mapped into the same ring as the RingElem.

Let r be a non-const 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)

Attempting to compute a gcd or lcm in a ring which not an effective GCD domain will produce an exception with code ERR::NotTrueGCDDomain. If r1 or r2 is a BigRat then an error is signalled at compile time.


CoCoALib offers functions for querying various properties of RingElems, and about relationships between RingElems.

Let r1 and r2 be a (possibly const) RingElems, and let N be a variable of type BigInt, and q a variable of type BigRat

Note that IsDivisible tests divisibility in the ring containing the values: so 1 is not divisible by 2 in RingZZ, but their images in RingQQ are divisible.


If the ring is an ordered domain then these functions may also be used. You can discover whether CoCoALib thinks that the ring R is arithmetically ordered by calling IsOrderedDomain(R): the value is true iff R is arithmetically ordered.

Note that comparison operations between RingElems will fail if they do not belong to the same ring (the exception has code ERR::MixedRings). You may perform comparisons between a RingElem and an integer or a rational -- the integer/rational is automatically mapped into the same ring as the RingElem.

Let r1 and r2 belong to an ordered ring. Trying to use any of these functions on elements belonging to a ring which is not ordered will produce an exception with code ERR::NotOrdDomain.

More operations on RingElems of a finite field

If owner(r) is a finite field, or a polynomial ring whose coeffs are ini a finite field then the following functions may be used. The P in the names of these function refers to the characteristic of the ring.

More operations on RingElems of a FractionField

If owner(r) is a fraction field then the following functions may be used. You can find out whether CoCoALib thinks that the ring R is a fraction field by calling IsFractionField(R): the result is true iff R is a fraction field.

Let K denote a FractionField Let r denote an element of K.

Note: the numerator and denominator are defined only upto multiples by a unit: so it is (theoretically) possible to have two elements of FrF which are equal but which have different numerators and denominators, for instance, (x-1)/(x-2) = (1-x)/(2-x)

More operations on RingElems of a QuotientRing

If owner(r) is a quotient ring then the following function may be called. You can find out whether CoCoALib thinks that the ring R is a quotient ring by calling IsQuotientRing(R): the result is true iff R is a quotient ring.

In addition to the standard RingElem operations, elements of a QuotientRing may be used in other functions.

Let RmodI denote a quotient ring. Let r denote a non-const element of RmodI.

More operations on RingElems of a RingTwinFloat

You can determine if an element belongs to a twin-float ring by calling IsRingTwinFloat(owner(r)): this yields true iff r belongs to a twin-float ring.

Let x, y be RingElem belonging to a RingTwinFloat

In contrast the test x==y may throw a RingTwinFloat::InsufficientPrecision while IsPracticallyEqual will never throw this exception. IsPracticallyEqual is intended for use in a termination criterion for an iterative approximation algorithm (e.g. see test-RingTwinFloat4.C).

More operations on RingElems of a PolyRing

You can determine whether an element belongs to a PolyRing by calling IsPolyRing(owner(r)): the result is true iff r belongs to a poly ring.

Let P denote a polynomial ring. Let f denote a non-const element of P. Let f1, f2 denote const elements of P. Let v denote a const vector of elements of P.

NOTE: to compute the weighted degree of a polynomial use the function wdeg defined for RingElem of a SparsePolyRing (see below).

More operations on RingElems of a SparsePolyRing

You can determine whether an element belongs to a sparse poly ring by calling IsSparsePolyRing(owner(r)): the result is true iff r belongs to a poly ring.

Let P denote a SparsePolyRing. Let f denote a non-const element of P. Let f1, f2 denote const elements of P. Let expv be a vector<long> of size equal to the number of indeterminates.

Let X be an indet (i.e. a RingElem in P) or a vector of indices (vector<long>)

NB For running through the summands (or terms) of a polynomial use SparsePolyIters (see SparsePolyRing).

We have still doubts on the usefulness of these two functions:

Use the following two functions with great care: they throw an error if the PPOrdering is not respected: (the coefficient c may be 0)

The corresponding member functions myPushFront/myPushBack will not check the validity of these assumpions: they should have a CoCoA_ASSERT to check in DEBUG mode.

More operations on RingElems of a DenseUPolyRing

You can determine whether an element belongs to a DenseUPolyRing by calling IsDenseUPolyRing(owner(r)): the result is true iff r belongs to a poly ring.

Let P denote a DenseUPolyRing. Let f denote an element of P.

WARNING Use this functions with great care: no checks on size and degree

Let f denote a non-const element of P.

Notes on operations

Operations combining elements of different rings will cause a run-time error.

In all functions involving two RingElems either r1 or r2 may be replaced by a machine integer, or by a big integer (an element of the class BigInt). 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 run-time error will be signalled if one attempts to compute a negative power of a non-invertible element. NB 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 run-time 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 read-only arguments which are RingElems, and RingElem& for read-write arguments. Unfortunately, doing so would lead to problems with the CoCoA library. INSTEAD you should use the types:

ConstRefRingElem x for read-only arguments: morally const RingElem& x
RingElem& x for read-write arguments
RingElem x for read-only arguments which make a local copy

If you are curious to know why this non-standard quirk has to be used, read on.

When accessing matrix elements or coefficients in a polynomial CoCoALib uses proxies: these are objects which should behave much like const RingElem values. To allow easy use of such proxies in functions which want a read-only RingElem we use the type ConstRefRingElem (which is actually const RingElemAlias&) for the formal parameter.

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 pointed-to 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 have called ConstRefRingElem.


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.

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 run-time 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 non-debugging 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 non-const raw value (e.g. raw(x), with x a RingElem), and r1, r2 potentially const raw values. Assume they are all owned by the ring R. Then the functions available are:

Maintainer documentation

(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 run-time 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 run-time, 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
RingElemAlias reference to a RingElem belonging to someone else
ConstRefRingElem C++ const-reference to a RingElemAlias
RingElemConstRawPtr raw pointer to a const ring value
RingElemRawPtr raw pointer to a ring value

For the first two see ring.

The classes RingElem and RingElemAlias are related by inheritance: they are very similar but differ in one important way. The base class RingElemAlias defines the data members which are inherited by RingElem. The essential difference is that a RingElem owns the value whereas a RingElemAlias does not. The two data members are myR and myRawValue: the first is the identity of ring to which the element belongs, and the second 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 differing ownership inherent in RingElemAlias and RingElem lead to several consequences. The destructor of a RingElem will destroy in the internal representation of the value; in contrast, the destructor of a RingElemAlias does nothing. A RingElemAlias object becomes meaningless (& dangerous) if the owner of the value it aliases is destroyed.

Why did I create RingElemAlias? 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. Analogous comments apply to the coefficients of a polynomial.

As already stated above, the internal data layouts for objects of types RingElem and RingElemAlias are identical -- this is guaranteed by the C++ inheritance mechanism. The subfield indicating the ring to which the value belongs is simply a ring, which is just a reference counting smart pointer. The subfield indicating the value is a raw pointer of type void*; 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*.

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 RingElemConstRawPtr::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.

Further comments about implementation aspects of the above classes.

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. RingZZImpl or RingFpImpl). These member functions follow certain conventions:

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.

ring element values are passed as raw pointers (i.e. a wrapped void* pointing to the actual value). A read-only 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.

sanity checks on the arguments are NOT CONDUCTED (e.g. the division function assumes the divisor is non-zero). These member functions are supposed to be fast rather than safe.

In a few cases there are non-pure 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:

Recall that arithmetic operations on objects of type ConstRefRingElem (which matches RingElem too) 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 read-only, and the return type is RingElem since it is new self-owning value (it does not refer to a value belonging 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 no-throw guarantee (according to the definitions in Exceptional C++ by Sutter).

Bugs, Shortcomings and other ideas

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.

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 BigInt 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.

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 (noncopying) access to coefficients in a polynomial via an iterator, or access to matrix elements?

Would we want ++ and -- operators for RingElems???

Should (some of) the query functions return bool3 values? What about properties which are hard to determine?

How to generate random elements from a ring?

Anna thinks that NearestInteger could handle specially elements of RingZZ rather than doing the full wasteful computation. Not sure if the extra code and complication would really make a difference in practice.

gcd and lcm: there is no guarantee on sign/monic because it may be costly to compute and generally useless.

Main changes