Definition talk:Module Direct Product

There is

Module of All Mappings is Module

Module on Cartesian Product is Module

Definition:Free Module Indexed by Set (which is needed to reformulate linear independence etc in a more practical way, and to rigorously introduce polynomials without resorting to tedious calculations)

The three proofs do special cases of the fact that a direct product of modules is a module. Interestingly, none of them does the general case. I suggest to keep one proof article that does the general case; the proofs in those particular cases are very very very similar anyway. The specific definitions themselves can go to the definition namespace.--barto (talk) 16:46, 4 December 2016 (EST)

Do the general case and then, if you must, add the proofs of the special cases with reference to the general case. --prime mover (talk) 16:47, 4 December 2016 (EST)

Left Modules

There was a reason to specify left modules in this definition.

An aborted attempt was made to define a tensor product as a construct involving both a left module and a right module, both of which had not been properly defined here. The original author expressed irritation at my insistence on defining all the entities in full and complete detail, and abandoned the project amid a cloud of bad feeling.

It was ever the intention to go back and fill in and distinguish between left direct products and right direct products, and the material is at a sort of half way house.

Please be aware that your work may endanger our attempts to make subtle distinctions between all the various types of object in this general area of mathematics, so please tread carefully here. --prime mover (talk) 17:55, 4 December 2016 (EST)

Thanks for the remark. I was very conscious when I removed that. I thought we first need a proper structure containing the basic things about modules, which does not exist yet. Later, we can add details about left and right modules everywhere, but it this moment that would only complicate the structuring.--barto (talk) 01:51, 5 December 2016 (EST)

We already do have what you say we have. I am puzzled as to why you think it all needs rewriting. --prime mover (talk) 02:26, 5 December 2016 (EST)

I could not find the definition of general direct products, the general definition of dimension of a module and an article addressing its well-definedness, an article stating that morphisms from free modules can be defined using their basis, the correspondence between multilinear maps and linear maps to function spaces, etc. All that is really basic and essential to be able to write any serious article about modules. And despite that, there are plenty of definitions and proofs doing only special cases, that are essentially duplicates. And even then, the proofs provided, while correct, are always calculatory while they can be much simplified and conceptualised using e.g. universal properties. Again, all that is really basic. That's why I think it needs rewriting. I'm just surprised that that hasn't been done yet. --barto (talk) 02:45, 5 December 2016 (EST)

Surprised why? --prime mover (talk) 03:30, 5 December 2016 (EST)

While it is appreciated that this area of maths on $\mathsf{Pr} \infty \mathsf{fWiki}$ gets some long overdue attention, we ought to be careful to avoid idiosyncrasies. It would be preferable to have two or more sources that at least can be broadly considered as the citations behind all the work you're doing.

It should furthermore be stressed that universal algebra is quite lacking. I made an endeavour into category theory, but it is not up to par yet. Complicating factor is that the XyJax package we use to render the commutative diagrams is out of active maintenance.

In any case, please make sure that, as much as reasonably possible, suitable definitions and concepts are defined or at least adequately red-linked and kept on some maintenance list (ideally somewhere in your user space). This makes sure that we avoid conflicting usage of the same concept by separate editors. The problems this caused in the model-theoretic realm have been around for five years and more, which severely limits the ease of extending that area. I would prefer if we can avoid this situation in the realm of universal algebra/commutative algebra/category theory.

In summary, please make sure that you have some overarching plan and apply it consistently, backed by at least one, but preferably multiple, published sources. — Lord_Farin (talk) 16:57, 5 December 2016 (EST)

Thanks for the suggestions. I'll make a todo-list (which I already have in mind) and provide a structure to be fleshed out, as you suggest. However, I will never find the courage to integrally copy books to Proofwiki as a reference, as I've noticed is being done. What I do is writing things I've learned at the universities where I study or have studied, at Wikipedia or other non-published sources... (By the way, as regards the book copying, I'm not in favor of such practice when it comes to abstract mathematics, as writing about something requires deep understanding and not just having read its definition. One has to understand the use of a concept in order to write and construct it in a proper way, otherwise we end up with lots of proofs and definitions that have little practical use, e.g. because they are too narrow cases of a more general thing that is useful. Or instead of proving something using a universal property, contributors give proofs by direct verification, which can be very tedious and not very instructive. While the effort is much appreciated, time is better spend on other things.) --barto (talk) 17:30, 5 December 2016 (EST)

We aim for the same thing. Of course it is not a good idea to mindlessly copy a book without understanding what is in there. On the other hand a printed source can help to structure presentation and to make sure certain pitfalls are avoided.

Another thing I'd like to draw attention to is that we ought to make sure that the definition of e.g. a polynomial, while constructed and described in full generality, is also still accessible to anyone with mere high school knowledge. This requires a deep understanding of the topic and a skillful combination of generality and specificity. So there is virtue to the tedious computation, if only to serve as a motivation to learn the more abstract mathematics that can provide the one-line proof.

In summary: we want the same, just please keep in mind that the less abstract views on certain topics do not become invalid or irrelevant when there is a more generic description available -- we all learned to integrate before we really got to know the meaning of $\mathrm dx$ as a differential. — Lord_Farin (talk) 12:22, 6 December 2016 (EST)