use Parent in quotient rings too

[fchapoton]

instead of using the deprecated ParentWithGens

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[fchapoton]

strange:

sage: FiniteDimensionalAlgebra(k,[],assume_associative=True)
Finite-dimensional algebra of degree 0 over Finite Field of size 5

is not in Rings()

and also

sage: C = AlgebrasWithBasis(QQ).example()
sage: C in Rings()
True

sage: A=Algebras(QQ)
sage: A.super_categories()
[Category of rings,
 Category of associative algebras over Rational Field,
 Category of unital algebras over Rational Field]

[fchapoton]

and now we have that

sage: from sage.categories.magmatic_algebras import *
sage: cat1 = MagmaticAlgebras(GF(3)).FiniteDimensional().WithBasis()
sage: cat2 = Algebras(GF(3)).FiniteDimensional().WithBasis()
sage: cat1.Associative()
Category of finite dimensional associative algebras with basis over Finite Field of size 3
sage: cat1.Associative() is cat2
False
sage: cat2
Category of finite dimensional algebras with basis over Finite Field of size 3

and simplified problem is

sage: cat1 = MagmaticAlgebras(GF(3))
sage: cat2 = Algebras(GF(3))
sage: cat1.Associative() is cat2
False

seems to be a difference in Unital

sage: cat3=cat1.Associative()
sage: cat3
Category of associative algebras over Finite Field of size 3
sage: cat3.axioms()
frozenset({'AdditiveAssociative',
           'AdditiveCommutative',
           'AdditiveInverse',
           'AdditiveUnital',
           'Associative',
           'Distributive'})
sage: cat2.axioms()
frozenset({'AdditiveAssociative',
           'AdditiveCommutative',
           'AdditiveInverse',
           'AdditiveUnital',
           'Associative',
           'Distributive',
           'Unital'})
sage: cat3.Unital() is cat2
True

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[fchapoton]

I have now made a simple first step in #38375.

[mantepse]

The only remaining bug is

file "src/sage/rings/finite_rings/finite_field_constructor.py", line 363, in sage.rings.finite_rings.finite_field_constructor.FiniteFieldFactory

right?

[mantepse]

And the reason for that failure seems to be that, in the implementation of QuotientRing_nc.gens, we have

    def gens(self) -> tuple:
        return tuple(self(self.__R.gen(i))
                     for i in range(self.cover_ring().ngens()))

and in the case at hand, i.e., k = GF(5, modulus='primitive'), the ring k.__R is the integer ring, with generator 1.

Note that FiniteField_prime_modn.gen has its own implementation, so maybe we should simply implement FiniteField_prime_modn.gens, too?

[mantepse]

No?