`coefficient_ring` for modules

[thofma]

If I remember correctly, the fanbase for base_ring is rather small (mainly because we used it in a different way than it was designed for) and we started introducing mathematically well-defined terms like coefficient_ring for polynomial rings.

I think we should do the same for modules and introduce coefficient_ring. So whenever we have a ring R and M is an R-module (does not matter which type it has or how it is constructed), I propose to have coefficient_ring(M) == R.

Any concerns? @wdecker @fingolfin @fieker?

[ederc]

I'm not sure I understand correctly but wouldn't it be more natural to have

base_ring(M) == R and coefficient_ring(M) == coefficient_ring(R) ?

[wdecker]

If R is a polynomial ring with coefficients in C, and M is an R-module, then M is an R-module but also a C-algebra. But still we are concerned with the R-module structure of M when we talk about modules. For example, if we have elements of M, we talk about R-linear combinations of these elements, or about linear combinations of these elements with coefficients in R. So coefficient_ring(M) == R seems to be fine. If we would consider instead an OSCAR type for C-algebras, what we don't, then ...

[ederc]

I see, alright.

[wdecker]

By the way, what I wrote fits also to the coefficient function for module elements:

julia> R, (x, y, z) = PolynomialRing(QQ, ["x", "y", "z"]) (Multivariate Polynomial Ring in x, y, z over Rational Field, fmpq_mpoly[x, y, z])

julia> A = R[x; y] [x] [y]

julia> B = R[x^2; y^3; z^4] [x^2] [y^3] [z^4]

julia> M = SubQuo(A, B) Subquotient of Submodule with 2 generators 1 -> xe[1] 2 -> ye[1] by Submodule with 3 generators 1 -> x^2e[1] 2 -> y^3e[1] 3 -> z^4*e[1]

julia> coefficients(x^2*M[1]) Sparse row with positions [1] and values fmpq_mpoly[x^2]

[fingolfin]

We just discussed this here. There weren't many concerns with the suggestion coefficient_ring(M) == R, but still some:

1. it does sound somewhat weird (but we didn't really have a better suggestion either...)
2. some of us found it confusing that if R = k[x,y] then for an ideal $I$ of $R$ we have coefficient_ring(I) == k while for an $R$-module we have coefficient_ring(M) == R. Of course ideal are not of type module, so one can justify that (and Claus thinks that's enough), but some of us (e.g. me) are still not really happy with this...

[fingolfin]

In the end, it seems it would be good to look at explicit usage cases to determine what is appropriate...