Definition talk:Tensor Product of Modules

I don't understand the last thing that is being asked for EmperorZelos (talk) 17:54, 24 November 2015 (UTC)


 * What is $-$ in the expressions $\left({m + m', n}\right) - \left({m, n}\right) - \left({m', n}\right)$ and so on? It is fairly clear that e.g. $\left({m + m', n}\right)$ is an ordered pair, i.e. an element of a Cartesian product of modules, but a) it is not clear what the subtraction operation is on these elements, and b) the subtraction operation needs to be explicitly defined on exactly what it means on an ordered pair. That's to start with.


 * I recommend that the notation used to define modules in this context be expanded into the full form $\left({G, +_G, \circ}\right)_R$ with the group operation defined as $\left({G, +_G}\right)$ and the ring operations specified uniquely as $\left({R, +_R, \times_R}\right)$, and a different notation devised to distinguish between a left module and a right module.


 * As it is, there is too much assumed knowledge and unexplained notational conventions used. I haven't read the texts, but I am prepared to bet dollars to dimes that the source works used (Lang and Bourbaki) take considerable pains to explain all the notation precisely.


 * Bottom line is: if you can't explain it, you don't understand it.


 * Remember when, at the start of this exercise, I said "I have to see any exposition on the subject that has even been able to explain worth a damn what a tensor is"? I still stand by that statement. --prime mover (talk) 21:32, 24 November 2015 (UTC)

Fair enough, I'll get on it EmperorZelos (talk) 06:44, 25 November 2015 (UTC)

Edition
how do I add chapter/page/edition to the book reference? EmperorZelos (talk) 16:00, 26 November 2015 (UTC)


 * I suggest you find examples. There should be plenty. --prime mover (talk) 19:42, 26 November 2015 (UTC)

Non-commutative ring
This part is a bit questionable. As far as I know, if the base ring $R$ is non-commutative, tensoring a right-$R$-module $M$ with a left-$R$-module $N$ naturally gives just an abelian group, without some nice $R$ module structure. --Z423x5c6 (talk) 06:59, 10 June 2021 (UTC)