Definition talk:Module Direct Product

There is

Module of All Mappings

Module Product

Module on Cartesian Product

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)