#16247: Meaning of Modules(R) currently not very clear
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Reporter: darij | Owner:
Type: defect | Status: new
Priority: major | Milestone: sage-6.2
Component: algebra | Resolution:
Keywords: modules, categories, | Merged in:
associativity, matrices | Reviewers:
Authors: | Work issues:
Report Upstream: N/A | Commit:
Branch: | Stopgaps:
Dependencies: #10963 |
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Changes (by darij):
* dependencies: 10963 => #10963
Old description:
> We have `LeftModules` and `RightModules` functorial constructions, and
> they should be used. `Modules` implements left and right multiplication
> to be *the same*, which causes misleading and counterintuitive non-
> associativity issues.
>
> 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. They should be bimodules! The reason why this doesn't blow up in
> the user's face (well, as far as I can tell) is that (I guess) the matrix
> space classes override the `*` operator to do the right thing (oops!)
> instead of use the defaults from the `Modules` category.
>
> 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.
New description:
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.
--
--
Ticket URL: <http://trac.sagemath.org/ticket/16247#comment:5>
Sage <http://www.sagemath.org>
Sage: Creating a Viable Open Source Alternative to Magma, Maple, Mathematica,
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