Comparison of polynomials from different rings throws inconsistently

[lkastner]

Describe the bug
When comparing polynomials from different rings I get inconsistent behaviour. Sometimes I just get false and sometimes an error.

To Reproduce

julia> R0,(x0) = polynomial_ring(QQ,1)
(Multivariate polynomial ring in 1 variable over QQ, QQMPolyRingElem[x1])

julia> R1,(x1) = polynomial_ring(ZZ,1)
(Multivariate polynomial ring in 1 variable over ZZ, ZZMPolyRingElem[x1])

julia> R2,(x2) = polynomial_ring(GF(2),1)
(Multivariate polynomial ring in 1 variable over GF(2), FqMPolyRingElem[x1])

julia> x0 == x1
false

julia> x0 == x2
false

julia> R3,(x3) = polynomial_ring(GF(3),1)
(Multivariate polynomial ring in 1 variable over GF(3), FqMPolyRingElem[x1])

julia> x2 == x3
ERROR: parents do not match
Stacktrace:
 [1] error(s::String)
   @ Base ./error.jl:35
 [2] check_parent
   @ ~/.julia/packages/AbstractAlgebra/QjFtH/src/AbstractAlgebra.jl:208 [inlined]
 [3] check_parent
   @ ~/.julia/packages/AbstractAlgebra/QjFtH/src/AbstractAlgebra.jl:207 [inlined]
 [4] ==(a::FqMPolyRingElem, b::FqMPolyRingElem)
   @ Nemo ~/.julia/packages/Nemo/4P3uO/src/flint/fq_default_mpoly.jl:295
 [5] ==(A::Vector{FqMPolyRingElem}, B::Vector{FqMPolyRingElem})
   @ Base ./abstractarray.jl:2943
 [6] top-level scope
   @ REPL[18]:1

julia> x0 == x3
false

Expected behavior
We have many places where we first compare the "surrounding space" before comparing the actual objects. So I would expect an error to get thrown. E.g. try

cube(2) == cube(3)

Additional context
Maybe this needs a styleguide entry, I actually did not look it up.

[lgoettgens]

Some background: In the cases with an error printed, the types of both arguments are equal, but the parents are not.
In the other cases, the types are not equal and thus julia delegated == to ===.

[lgoettgens]

Let's discuss the possible fix in Nemocas/AbstractAlgebra.jl#1800 in triage.

[mjrodgers]

It seems that there are cases where we would want an error rather than false when the parents are not ===. So maybe this inconsistency is hard to avoid. After discussing in triage, no conclusion on proper behavior was reached.

[mjrodgers]

arguably, always throwing an error might be preferable.

[fingolfin]

In my personal experience (which of course may differ from what others experience) is that if I compare two polynomials from different rings, then this is always a user error (i.e., mine). In that situation a helpful error message that suggests what's wrong for me is preferable to getting a result that may confuse me (i.e. think of f and g both printing as x but f == g giving false and the user going "WTF").

That said, perhaps people have concrete examples in mind were it is helpful to be allowed to compared polynomials with different parents directly. In that case it'd be great to learn about them here.