Definition talk:Right Module

From ProofWiki
Jump to navigation Jump to search

The definition of a right module is a $R$-algebraic structure which has a binary operation $\circ: R \times G \to G$ but the right module axioms are written using infix Notation suggesting that the binary operation is actually $\circ: G \times R \to G$ ($G$ and $R$ are reversed). This seems to me to be inconsistent and I think it masks an important aspect of what is being defined in the definition of a module which is both a right module and a left module.

There a two possible ways to resolve this:

(1) Recast the right module axioms in terms of the binary operation $\circ: R \times G \to G$.
\((RM \, 1)\)   $:$   Scalar Multiplication Right Distributes over Module Addition      \(\displaystyle \forall \lambda \in R: \forall x, y \in G:\)    \(\displaystyle \lambda \circ \paren {x +_G y} \)   \(\displaystyle = \)   \(\displaystyle \paren {\lambda \circ x} +_G \paren {\lambda \circ y} \)             
\((RM \, 2)\)   $:$   Scalar Multiplication Left Distributes over Scalar Addition      \(\displaystyle \forall \lambda, \mu \in R: \forall x \in G:\)    \(\displaystyle \paren {\lambda +_R \mu} \circ x \)   \(\displaystyle = \)   \(\displaystyle \paren {\lambda \circ x} +_G \paren {\mu \circ x} \)             
\((RM \, 3)\)   $:$   Associativity of Scalar Multiplication      \(\displaystyle \forall \lambda, \mu \in R: \forall x \in G:\)    \(\displaystyle \paren {\lambda \times_R \mu} \circ x \)   \(\displaystyle = \)   \(\displaystyle \mu \circ \paren {\lambda \circ x} \)             

This approach would require some explanation of the axiom ($RM3$) explaining that it is derived from a notion of applying the action on the right. It would also have the least impact on the definition of a module and would make it easier to highlight the salient feature of a (two-sided) module, namely that the following equalities hold:

$\lambda \circ \paren {\mu \circ x} = \paren {\lambda \times_R \mu} \circ x = \mu \circ \paren {\lambda \circ x} = \paren {\mu \times_R \lambda} \circ x$

Essentially the actions are commutative even if the ring is not commuatative.

(2) Introduce a right $R$-algebraic structure and define a right module in terms of this. This would make the definition of a module a little more complicated where:
A module over $R$ is an $R$-algebraic structure with one operation $\struct {G, +_G, \circ}_R$ such that:
$\struct {G, +_G, \circ}_R$ is a left module
$\struct {G, +_G, \tilde \circ}_R$ is a right module
where $\tilde \circ: G \times R \to G$ is the binary operation defined by $x \tilde \circ \lambda = \lambda \circ x$.

In either case all of the pages dependent on these pages would need to be reviewed and made to fit with the changes. But I think all of the pages need to be reviewed as there are a number of inconsistencies existing that are the result of a module initially being left modules and then being changed to also be right modules. For instance:

(a) Many of the theorems in Category:Module Theory only prove the axioms for a left module and never for a right module. While the axioms for a right module may follow trivially they are not even given a mention. Most of the theorems in Category:Module Theory have an equivalent theorem for left modules and right modules but these are not stated anywhere.
(b) The definition of a vector space is that it is a left module, but in one instance (and I'm sure there are more) a proof makes use of a theorem about modules: Homomorphic Image of Vector Space appeals to Homomorphic Image of R-Module is R-Module to conclude that the image is a $K$-module. But the vector space is only assumed to be a left module.

I'm looking to start reworking these pages in my sandbox, but before I do this I would be interested in others thoughts on which approach I should take. I should also note that option (1) above is not an approach that I have seen in any source, although there is no reason it wouldn't work. Most books take an approach like option (2). I only offer option (1) as it seems to be more aligned to what is already existing on $\mathsf{Pr} \infty \mathsf{fWiki}$.

--Leigh.Samphier (talk) 09:38, 25 July 2019 (EDT)

The initial work on modules was done with reference to left modules only (source work was Warner). I reached a point at which I started to get bogged down, so I abandoned it as it was well outside the work I did for my formal studies.
Then a couple of other people picked it up and seemed to know what they were doing, but left the material in an inconsistent state which needed to be tidied up. I always intended to return to it but never did.
I don't understand what the problem is. The way I see it is that in a left module, the scalars are on the left, and in a right module, they are on the right, so indeed the binary operation is actually $\circ: G \times R \to G$. This is how I intuitively understand it, and I thought it was a clear and approachable exposition (except that the actual binary operations $\circ: G \times R \to G$ and $\circ: R \times G \to G$ were not explicitly pointed out).
As I say, I don't understand the concerns, but then I have not found any discussion of right modules in the literature so I can't really help. (Maybe I have some such materials in my personal library, but I would need to hunt for it.
I don't understand approach 1. It seems that a right module is being defined in terms of scalars on the left, which does not make intuitive sense -- and if this is the case, then distinguishing between a right and a left module operation would seem to be a problem. (What if you have a structure in which you have defined both a left module and a right module with respect to a group and a scalar ring, and they are different, and you need to distinguish between the two?) As for approach 2, apart from the fact that the notation is clumsy and arbitrary (I don't like using primes and tildes to distinguish between two dual structures, it looks only half thought through), but if this is standard, then that sounds okay.
But what I need to know is: what is actually wrong with the current approach which (as I say) is understandable? --prime mover (talk) 11:02, 25 July 2019 (EDT)
So someone else is going to need to comment.
I guess the first issue for me is that for a right module the operation is defined to be $\circ: R \times G \to G$ and not $\circ: G \times R \to G$, because this is the way an $R$-algebric structure is defined. By the definitions in place, the scalar operation should be on the left when using infix notation, yet the axioms have the scalar operation on the right. There is no explanation for this switch and why it is allowed. If I have a noncommutative ring and a two-sided ideal, I can form a right module, but there are two right operations possible: the mirror of the left operation or the natural restricted operation on the right, and these operations are different because of the noncommutativity. I think a right module should be defined as a operation $\circ: G \times R \to G$.
The second issue for me is that of a module. The definition has a single operation $\circ:R \times G \to G$ such that it is both a left module and a right module. If $R$ and $G$ are not equal, then a binary operation can't be both a mapping from $R \times G$ and a mapping from $G \times R$. There has to be two operations, and the relationship between the operations explained. --Leigh.Samphier (talk) 17:38, 25 July 2019 (EDT)
"because this is the way an $R$-algebric structure is defined." Again, this is how it is on $\mathsf{Pr} \infty \mathsf{fWiki}$ because this is how it is defined in the only source that has been used to define it. Warner defines an $R$-algebraic structure with the scalar on the left. Perhaps this should be called a "left $R$-algebraic structure", and the one with the scalar on the right a "right $R$-algebraic structure". The whole concept is arbitrary and synthetic anyway. Warner focused on left modules only. Maybe other sources contain this concept ... and in fact when I pick up 1989: P.M. Cohn: Algebra: Volume $\text { 2 }$ (2nd ed.) (which I haven't really looked at properly yet, apologies) this is exactly the approach that is taken.
What sources are you using? --prime mover (talk) 18:09, 25 July 2019 (EDT)