Where do polynomial rings get their scalar multiplication action?

[Nils Bruin]

On https://github.com/sagemath/sage/pull/39371 I found that R['x'] knows how to find the scalar multiplication action of R on R['x'] in most cases, but that for R=Qp this is not the case. It looks like there are some benefits from relying on such an action, so it would be nice if this action were discoverable in all cases. My questions:

 - where does this action get installed for most rings R?

 - is there a reason why for Qp this action is not installed?

[Nils Bruin]

It seems that padic polynomials simply have a malfunctioning _rmul_ !!!

    def _rmul_(self, left):
        return None
        # The code below has never been tested and is somehow subtly broken.

see:

https://github.com/sagemath/sage/blob/5188024881df509aee12e59e4d8ebf003c675e4a/src/sage/rings/polynomial/padics/polynomial_padic_capped_relative_dense.py#L576-L577

it shouldn't be too hard to fix that? Someone with some experience with the code there could probably fix that quite easily.