Unify `residue_ring` vs. `quo`

[joschmitt]

We have residue_ring and we have quo (and residue_field?) and they don't always do the same.
A real issue is the behaviour exposed in #4410 that residue_ring always (silently) assumes that the ring is euclidean. At least for multivariate polynomial rings, we can do better here by returning a proper multivariate quotient ring (this only works in OSCAR and not in AbstractAlgebra of course).
I'm aware that part of the problem is that we can't easily test whether a ring is euclidean, but at least we could clean up the names. Or is there some distinction that I don't see?

To quote @JohnAAbbott in #4410:

In any case, it is not good that residue_ring returns a EuclideanRingResidueRing when the result clearly is not -- it should either produce an error or give a correct result. I did not see anything in the documentation about restrictions on arguments for residue_ring.

[JohnAAbbott]

Here is a simple test case (taken from #4410

# using Pkg; Pkg.develop(path="/home/zez25zuh/OSCAR/GIT/AbstractAlgebra.jl");
# using AbstractAlgebra
P,(x,y) = polynomial_ring(QQ, ["x", "y"]);
f = x+y+1;
R,_ = residue_ring(P,f);
typeof(R)

This produces the incorrect result:

EuclideanRingResidueRing{AbstractAlgebra.Generic.MPoly{Rational{BigInt}}}

[fieker]

genrically, my view: quo should always exist and always return the projection. the others wildly differ. For many types and a long time, the map was non-existing (well there were no maps and lift and coerce did the job), so quo should cover everything and is stable.
residue_ring vs residue_field is for optimization and type stability. Other features:

 julia> QQx, xx = QQ[:x]
(Univariate polynomial ring in x over QQ, x)

julia> residue_field(QQx, xx^2-2)
(Number field of degree 2 over QQ, Map: QQx -> number field)

julia> residue_ring(QQx, xx^2-2)
(Residue ring of QQx modulo x^2 - 2, Map: QQx -> residue ring)

julia> quo(QQx, xx^2-2)
(Residue ring of QQx modulo x^2 - 2, Map: QQx -> residue ring)

julia> quo(Field, QQx, xx^2-2)
(Number field of degree 2 over QQ, Map: QQx -> number field)

[simonbrandhorst]

genrically, my view: quo should always exist and always return the projection. the others wildly differ. For many types and a long time, the map was non-existing (well there were no maps and lift and coerce did the job), so quo should cover everything and is stable. residue_ring vs residue_field is for optimization and type stability. Other features:

 julia> QQx, xx = QQ[:x]
(Univariate polynomial ring in x over QQ, x)

julia> residue_field(QQx, xx^2-2)
(Number field of degree 2 over QQ, Map: QQx -> number field)

julia> residue_ring(QQx, xx^2-2)
(Residue ring of QQx modulo x^2 - 2, Map: QQx -> residue ring)

julia> quo(QQx, xx^2-2)
(Residue ring of QQx modulo x^2 - 2, Map: QQx -> residue ring)

julia> quo(Field, QQx, xx^2-2)
(Number field of degree 2 over QQ, Map: QQx -> number field)

When I was new to Oscar the difference between residue_ring and quo confused me a bit. Also I wondered why there is no quotient_ring. I figured that residue_ring was for number theory only and ignored it henceforth. I kept using residue_field. I never even suspected that quo(Field,QQx,xx^2-2) even exists. But it seems to be a more common pattern nowadays, so I can see the point.

My gut feeling is that having a sometimes equivalent and sometimes broken equivalent of quo for rings is more confusing than useful.

Why is residue_ring better for optimization and type stability than quo (or do you mean to keep it for euclidean rings only? But maybe there are some methods of residue_ring for non euclidean rings already?) ?

Aside there is also

   quotient(G::MultTableGroup, H::MultTableGroup, HtoG::MultTableGroupHom)

  Returns the quotient group Q = G/H with canonical map G -> Q.

which could/should be quo instead?