On Wed, Sep 07, 2011 at 02:27:47PM -0700, William Stein wrote:
> > Moreover, one must not forget that we want to create quotient rings
> > not only for ideals in polynomial rings.
>
> True, but it is an important case, since the ideas covers all finitely
> generated quotient rings, and there are a lot of finitely generated
> rings in Sage.
To be precise, finitely generated *commutative* quotient rings :-)
My point is just that the ideals/quotients framework will be used in a
fairly large context. So it would be good if it did not impose too
many constraints, like forcing '==' to *always* mean mathematical
equality (though it could in the simple/not too expensive cases).
Otherwise, this will prevent modeling complex objects where some
lazyness is vital.
For a concrete example, take a quotient of a free non commutative
algebra by a graded ideal. Its Gröbner basis will typically be
infinite, so one can't compute it in full. Yet, arithmetic can be
achieved by lazily computing it up to the appropriate degree.
In general, after three years, I still remain a big fan of MuPAD's
approach where ``=='' meant syntactical equality, and mathematical
equality was achieved with a different syntax. Especially since ``==''
is used implicitly in many situations (coercion, arithmetic,
dictionary) that ought to be super fast.
Cheers,
Nicolas
--
Nicolas M. Thiéry "Isil" <nthi...@users.sf.net>
http://Nicolas.Thiery.name/
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