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.
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");
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.
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.
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.
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
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.
While considering the design of this code it may help to keep in mind these two canonical implementations:
GeneralQuotientRingImpl
RingFpImpl
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.
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.