Meaning of Modules(R) currently not very clear

[darijgr]

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.

Depends on #10963

CC: @nthiery @simon-king-jena @orlitzky

Component: algebra

Keywords: modules, associativity, matrices

Issue created by migration from https://trac.sagemath.org/ticket/16247

[darijgr]

Changed keywords from modules, associativity to modules, associativity, matrices

[darijgr]

Dependencies: 10963

[nthiery]

comment:4

Thanks for exploring how to clean this up.

Just for the record: we have had left/right/bimodules since 2009 and probably even before. Also, at this point, they are just categories over a base ring, not functorial constructions.

One thing to check is how this was handled in Axiom. MuPAD was roughly as here: on the lousy side in the non commutative case.

[darijgr]

Changed dependencies from 10963 to #10963

[mkoeppe]

comment:9

Setting new milestone based on a cursory review of ticket status, priority, and last modification date.

[pjbruin]

comment:11

I got here through #32250, in which I propose a precise definition for (magmatic) algebras over an associative unital ring, and #32250 comment:2.

My personal preferred solution would be a variant of the first option in the ticket description: make Modules only support commutative base rings, and define a module over a commutative ring R to be a symmetric (R, R)-bimodule. This feels most in line with how modules are currently used in Sage. Supporting non-commutative rings with this definition would probably be more confusing than justified by the little amount of added functionality.

[orlitzky]

comment:12

Replying to @pjbruin:

My personal preferred solution would be a variant of the first option in the ticket description: make Modules only support commutative base rings, and define a module over a commutative ring R to be a symmetric (R, R)-bimodule. This feels most in line with how modules are currently used in Sage. Supporting non-commutative rings with this definition would probably be more confusing than justified by the little amount of added functionality.

I agree. This is what the existing Modules? docstring is trying to say. The cited UserWarning more or less says that your ring should be commutative, and that your left- and right-module actions had better be the same. The definition should match what Sage is actually capable of.

This would leave us with new problem though, that of defining a symmetric bimodule within sage. Maybe we should just be explicit about the axioms instead of using that term.