either characteristic or equality test in localizations of quotients is lying

[thofma]

@HechtiDerLachs For the ring

julia> Qxyz, (x, y, z) = QQ["x", "y", "z"];

julia> I = ideal(Qxyz, x^2 - x, x * y);

julia> Q, = quo(Qxyz, I);

julia> L, = localization(Q, powers_of_element(x) * powers_of_element(x - 1));

I get

julia> iszero(one(L))
true

julia> characteristic(L)
0

Any idea what could be wrong?

P.S.: I got this from an example in the geometry tests, while looking for usages of the zero ring.

[thofma]

I guess the multiplicative set contains zero, so the characteristic is wrong.

[fingolfin]

yeah, this method is overly optimistic:

characteristic(W::AbsLocalizedRing) = characteristic(base_ring(W))

[thofma]

Fixing it means that we will see that during the Oscar tests, a multivariate polynomial ring over a genuine zero ring is constructed. More precisely, this happens in the fibre_product function (see https://github.com/Nemocas/AbstractAlgebra.jl/actions/runs/11898952988/job/33156584094?pr=1857#step:7:3299). Since we now know that the resulting polynomial ring is bugged:

julia> L, = localization(Q, powers_of_element(x) * powers_of_element(x - 1));

julia> L, (x, y) = L[:x, :y];

julia> x == 1 * x
false

we need to discuss how to deal with this:

• Can the fibre product avoid zero rings?

[HechtiDerLachs]

Sorry for not getting back to this yet! I didn't have this on the radar when implementing characteristic, so thanks for pointing it out. I will look into this on Monday.

[afkafkafk13]

Stupid question: Shouldn't the localization know about an element of the modulus being inverted, when it creates the localized ring?
If so, shouldn't it 'mark' the localized ring as the zero ring in some way for later use?

[HechtiDerLachs]

Shouldn't the localization know about an element of the modulus being inverted, when it creates the localized ring?

I think, it does not make sense to have the constructor look for this, really. Why would we do this extra check?

If so, shouldn't it 'mark' the localized ring as the zero ring in some way for later use?

At any point one can check whether or not a ring is the zero ring and get the correct answer. If this can cheaply be cached, then one could do it, e.g. by setting an attribute. But even if one doesn't, one can still do the check with no harm.