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16247 Meaning of Modules(R) currently not very clear darij "The doc of `class Modules` currently (#10963) says:
{{{
The category of all modules over a base ring `R`.
An `R`-module `M` is a left and right `R`-module over a
commutative ring `R` such that:
.. math:: r*(x*s) = (r*x)*s \qquad \forall r,s \in R \text{ and } x \in M
}}}
This is not the notion of a module that mathematicians are used to, not even when R is commutative. Instead, this is the definition of an R-R-bimodule. I fear that this is destined to lead to confusion and subtle bugs. For instance, the `WithBasis` subcategory implements methods like ""basis"" and ""support"". But a left R-module basis of an R-R-bimodule might not be a right R-module basis, and even if it is, the supports of one and the same element with respect to it (one time as a left R-module basis, another time as a right one) might be different. I have not seen the `WithBasis` subcategory being used in problematic cases (i.e., in cases where the left and right structure are different), but I fear that this is bound to eventually happen.
I've run the (short) doctests of src/sage with a commit that adds a warning every time Modules(A) is called for A noncommutative. Here are the relevant results:
https://www.dropbox.com/s/oieg1ig0dliz63s/noncomm.txt
It seems that matrices over noncommutative rings are the main culprit here -- or, rather, matrix spaces being cast as modules over the base rings. I think they should be bimodules, since there is a `Bimodules(R, R)` category already.
Apparently people have been aware of this for a while; the following warning message is doctested for and not written by me:
{{{
doctest:...: UserWarning: You are constructing a free module
over a noncommutative ring. Sage does not have a concept
of left/right and both sided modules, so be careful.
It's also not guaranteed that all multiplications are
done from the right side.
}}}
(We do have left/right/bi-modules now.)
There are some tracebacks I don't really understand... can it be that some methods in Sage construct matrices consisting of matrices? There's nothing wrong about that; I just think the constructor for the respective matrix spaces should pick the right category for that.
Here are some options:
- Make `Modules` only support *symmetric* modules, i.e. modules M satisfying rx = xr for all r in R and x in M. This is useful almost only for commutative R (in fact, these modules are always modules over the abelianization of R).
- Make `Modules` only support R-R-bimodules which are direct sums of copies of the R-R-bimodule R. This allows for doing most things that can be done in the commutative case, and examples are polynomial rings over noncommutative rings, matrix spaces etc. -- I actually like this category. The only problem is that it is more of a ""ModulesWithBasis"" category than a ""Modules"" category.
- Make `Modules` only support R-R-bimodules which are sums (not necessarily direct) of copies of the R-R-bimodule R. This looks like a reasonable category but I know almost none of its properties." defect new major sage-9.3 algebra modules, categories, associativity, matrices nthiery SimonKing N/A #10963