Implement is_injective for ring homomorphisms

[saraedum]

Currently, is_injective is almost never implemented:

sage: ZZ.hom(QQ).is_injective()
NotImplementedError
sage: QQ.hom(QQ).is_injective()
NotImplementedError

This makes the method is_subring mostly useless since its default implementation relies on is_injective of the "natural map":

sage: GF(3).is_subring(GF(9))
NotImplementedError

The changes introduced here, implement it as a method of the morphisms in the category of Rings where we can handle many common cases of morphisms (domain is a field, ….)

Note that is_injective is not shown in the tab completion anymore. A problem that is fixed in #21880.

[saraedum]

New commits:

​aa5fd5c	fixed typo
​1a5fac4	Implement is_injective as a MorphismMethod

[git]

Branch pushed to git repo; I updated commit sha1. This was a forced push. New commits:

​6890281

Implement is_injective as a MorphismMethod

[saraedum]

Note that is_injective is now faster even for the one case where it just returned True before.

sage: K.<x> = FunctionField(QQ)
sage: f = K.hom(x)
sage: %timeit f.is_injective()

Went down from 182ns to 76ns (which is of course due to the @cached_method.) Without it, it goes up to 13µs.

[git]

Branch pushed to git repo; I updated commit sha1. New commits:

​e10a257	Cache is_injective
​363ece8	Remove redundant is_subring() implementations

[git]

Branch pushed to git repo; I updated commit sha1. New commits:

​2f641e2

Add missing doctest output

[saraedum]

New commits:

​2f641e2

Add missing doctest output

[roed]

Assuming that the patchbot is happy, I am.

[saraedum]

Actually, only one of #21894 and #21895 is a dependency. Either one makes the patchbot happy.

[vbraun]

You should just figure out what you actually want to depend on. This ticket will not be merged until all dependencies are merged.

[saraedum]

vbraun: I know. I will remove one of the dependencies as soon as the other one is merged.

[roed]

So, should the #21894 dependency be removed?

[saraedum]

Yes, I can replace it with a much easier one.

[saraedum]

We do not need the full #21894. #19820 is enough to pass the doctests. Alternatively, the revert of ​https://git.sagemath.org/sage.git/commit?id=86b4868e9deb41940e3bed2a3289276bd7557a52 would also be enough but I think that #19820 is a more fundamental fix to the underlying problem (broken __eq__ implementations.)