On Tuesday, June 27, 2017 at 4:31:44 AM UTC-5, Jeroen Demeyer wrote:
>
> Suppose that R is some parent which is contained in the category 
> Algebras(B) for some ring B, how does one determine B from the parent R? 
>
> I am asking because I want to implement a new method in the 
> ParentMethods of Algebras, which needs to refer to the base_ring of the 
> category. I feel like ParentMethods should have a way to refer back to 
> the category where the ParentMethods was defined. 
>

If I understand what you're asking correctly, you can just ask for the base 
ring of the category:

sage: Algebras(QQ).WithBasis().Graded()
Category of graded algebras with basis over Rational Field
sage: _.base_ring()
Rational Field

Although not tall categories are defined over the exact base ring of the 
algebra, but recall that sometimes instead they are given over a category 
like Rings or Fields(). This was done so we didn't have to create a number 
of categories over, say, GF(p) as the prime p ranges. So this is not always 
a safe method to get the base ring of the parent. However, I'm +1 for doing 
something to remove base_ring() from Parent. Unfortunately, this is 
probably not the way to do so.

Best,
Travis

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