Undergrad Advantages of Vector Spaces over Modules

[FallenApple]

I know that vector spaces have more structure as they are defined over fields and that modules are defined over rings. But it's hard to think of a situation where a using a ring clearly backfires. Is it just because a ring doesn't have an inverse for the second operation?

For a module over Z where addition has an inverse and multiplication doesn't, I could multiply a module element by an integer. Sure I might not be able to reverse it by division. But I could have just not multiplied in the first place. Or I could just keep track of where I came from by storing the previous location if I am using a computer, in that case it would just be more storage space.

 

[fresh_42]

I know that vector spaces have more structure as they are defined over fields and that modules are defined over rings.

Not sure if we should call this "more" structure. All vector spaces are modules but not vice versa, i.e. modules are the more general concept.

But it's hard to think of a situation where a using a ring clearly backfires. Is it just because a ring doesn't have an inverse for the second operation?

Well, one defines a nice and often smooth plane and the other one only a lattice. Many physical quantities can have continuous values, so the integers might not be enough. In QFT where quantized values play a much more important role, there is an entire section that uses lattice theory.

For a module over Z where addition has an inverse and multiplication doesn't, I could multiply a module element by an integer. Sure I might not be able to reverse it by division. But I could have just not multiplied in the first place. Or I could just keep track of where I came from by storing the previous location if I am using a computer, in that case it would just be more storage space.

Modules play a far greater role than you might think. E.g. every group or algebra representation is also a module, which includes semi-direct products. So modules are everywhere, it just happens that some of them are even vector spaces.

The way they are treated in textbooks is usually a different one, linear algebra vs. ring theory. This reflects one of the purposes why we consider them: In linear algebra we investigate the transformations between vector spaces while modules in connection with ring theory are used to find out properties of the rings. But you can always say $\mathbb{R}$ module to a real vector space.

 

[FallenApple]

Not sure if we should call this "more" structure. All vector spaces are modules but not vice versa, i.e. modules are the more general concept.

Well, one defines a nice and often smooth plane and the other one only a lattice. Many physical quantities can have continuous values, so the integers might not be enough. In QFT where quantized values play a much more important role, there is an entire section that uses lattice theory.

Modules play a far greater role than you might think. E.g. every group or algebra representation is also a module, which includes semi-direct products. So modules are everywhere, it just happens that some of them are even vector spaces.

The way they are treated in textbooks is usually a different one, linear algebra vs. ring theory. This reflects one of the purposes why we consider them: In linear algebra we investigate the transformations between vector spaces while modules in connection with ring theory are used to find out properties of the rings. But you can always say $\mathbb{R}$ module to a real vector space.

I think I need the smooth version over the lattice version.

Some context: I'm working on transforming the number line into a vector space using infinite basis that are isomorphic to the primes where the metric of this space measures the amount of mutual information they contain(which I so far have defined as the number of overlapping primes). The reason that I want to do is to apply statistical methods to see patterns, such as entropy of primes, which rely on vector spaces. I don't know just how strict this is.

What I ended up getting is a module over Z, which I think might not be sufficient because I think things like the Gauss-Markov theorem might not work on a space where the scalar multiplication isn't reversible.

Here is a number 60 in module form.

$60=3*5*2^{2} <=> log(60)=log(3)+log(5)+2*log(2)$ which in turn can be reexpressed as $|60>=|3>+|5>+|2>+|2>.$

The ket is supposed to mean vector, but I'll use it for module as well. Basically, any number in Z as a linear combination of primes transformed. And I don't need a notion of multiplication since I can just sum $|2>$ twice.

But if I want to actually express it nicely as $|60>=|3>+|5>+2*|2>$. I would need to have multiplication of a ring element with that group element $|2>$
I would need to define the space not as a abelian group under addition, but as a module over a Z ring.

If I want to extend the ring multiplication operation to have an inverse, I would need to have 2->1/2. But $\frac 1 2 |prime>$ Which is isomorphic to $\frac 1 2 log(prime)$. This doesn't have much of a meaning because I cannot express composite numbers as fractional powers of primes.

Yet I need to if I want to extend this module to a full vector space for statistical purposes.
I take from your post that I would need to fill in the gaps, since I should make it an R module? If I want to take the linear algebra approach.

So I need to somehow use fractional powers as scalars and have it correspond to something meaningful? Perhaps filling out the number line with rationals? What about irrationals? I could let multiplication to be reversible by defining an inverse. So roots can be expressed as $\frac 1 2 |prime>$. But it's hard to start from the abstract space and move towards the numbers. I can do $\pi$ as $|\pi>=a*|prime_{1}>+b* |prime_{2}>+c* |prime_{3}>+...$ Where the primes are in no particular order. This amounts to finding an infinite product of primes, each one risen to some power, such that it converges to be pi. Which is not feasible, sadly.

 

[fresh_42]

Well, I don't think that leads anywhere. If the primes are your basis vectors, you could restrict yourself to the rational number line instead of the real. Then you have to decide how to handle $\{0,\pm1\}$ and $\frac{1}{p}.$ However, it looks to me as if either you'll end up with linear dependent vectors (primes), or, in case you formally separate primes and scalars, it doesn't play any role, what the scalars are. For your ansatz it may be better to think of a domain that acts on your module, and to define the module operations first, than to begin with scalars in a field as the domain.

In my opinion you cannot avoid the question, what exactly do you want to describe? An eventual answer should automatically lead to a model. Another question is, have you looked on theorems about prime number and / or prime factor distributions? I suppose there are plenty of them, but I'm no number theorist. There are still open questions about primes, so it's hard to believe that it would be an easy task to find a new approach.

In order to understand entropy, it is probably better to consult a book about stochastic than to model it by primes.

 

[FallenApple]

So far, the operations that I have defined on my module is $+$and it's inverse$-$. And multiplication$*$. But the metric that I'm using uses roots, which technically doesn't exist within the defined operations, so maybe I need to truncate the negatives or get rid of the roots.

I can use $|Prime>=(0,0,...k*Prime,0,0,...)$ to separate the scalars and the primes. Though k can be prime, but it probably means there is some infinitely recursive automorphism going on where the k's that are prime can also be considered as basis in their own higher dimensional spaces. Though I just consider k as a scalar overall.

Even though it's most likely better to just go with the number line, something just feels more natural about expressing things in terms of primes. For example, in this hyperspace, composite numbers are just projections from higher dimensional space onto a subspace. $|2>$ can be expressed as a projection from $|18>$.
Where $|18>=|2>+2*|3>$ It's as if numbers as objects are just shadows of higher dimensional objects that encapsulate the lower dimensional form, in some sense.

For general k,
$|2>+k*|3>$ projects onto the subspace that $|2>$ exists in. By letting k=0, we get $|2>$. Which is a projection from a higher dimensional space.

Also, consider the number $3$. Ordinarily $3$ is closer to $27$ than it is to $99$. However, in this space, with metric $d(x,y)=\sqrt{<x-y,x-y>}$
$d(99,3)=\sqrt{2}$
$d(27,3)=2$

Which makes sense as 27 has more redundant information in its relation to 3 than 99 is in its relation to 3. Both 27 and 99 has two extra prime vectors than 3. But 99 is the one that has a new basis vector($|11>$) while 27 just has two extra repeated($|3>$) compared to 3. So there's a penalty for redundant information.

Since this is infinite dimensional and uses Euclidean distance, it should be a Hilbert Space, with the problem being that it is a so far a lattice, and not a complete space. Although the metric should still be informative.

Overall, a different type of similarity between numbers can be measured in a linear geometric way that is a bit different than the absolute distance on the 1d number line.

This brings me to a followup question. Do I have to define roots in order to use the metric? It would just be the distance between lattice points. I could allow rationals to be part of the domain. But $number=Prime1^{k1}*Prime2^{k2}...$ wouldn't be a prime factorization anymore in the general case. I can still use $|number>=(k1,k2,...)$ as a vector but k1 as fractional instead of as a whole number. I think the metric would still be meaningful here. Would allowing rational multiplicities of primes coordinates help fill the number line for each axes that correspond to each prime? It's like each axes is a number line, but represents multiplicities of particular primes instead of multiplicities of 1.
I'm not sure exactly yet if any of this can help in analyzing entropy. It's not crucial that I use this to solve entropy problems, it'd be great but I'm just approaching this opendedly and see where it leads me.

I did realize that in general, when things are linearized, the surfaces just fit better with optimization because Newton's method wouldn't get stuck at a local minimum or diverge to infinity. So if this group can be made complete, there might be a way to use it somehow by reverse transformation. This might be useful in some way if we use optimization techniques with with some sort of stochastic, maybe a variant of Monte Carlo, measure of frequency.