Module Over A Ring Puns

Been thinking this one quite a while but I cant find an answer. The ring has to be commutative.

EDIT: This can't be a Noetherian module , if this kind of module doesn't exist means that if in a module, every proper submodule is finitely generated, then the entire module is, thing which I don't think is true, but that is why I'm searching a counterexample. If this is true, please tell me where I can find a proof.

Thanks in advance.

I am just curious if anyone would be willing to discuss the general theory of modules over rings with a non-associative/alternative multiplicative group multiplicative structure. Any recommendations that would point me in the right direction would be fantastic (texts/papers/etc.). In particular, I'm interested in modules over the real, complexified, and quaternionified octonions. But other octonionic structures are more than welcome to the discussion, as well as modules over other non-associative rings.

Disclaimer: I have only been attacking abstract mathematics for a bit less than a year, but I am beginning graduate studies in the fall. So, my level of mathematical maturity is still rather low, and I must apologize if my inquiries are not entirely well-posed. I have a tendency to jump in over my head to keep myself moving forward, so certainly use whatever language is necessary for a proper discussion. That said, more explicit/gentle explanations would be greatly appreciated when possible.

Thanks!

With the current launcher capabilities why hasnt nasa launched an artificial gravity ring to the iss yet?

I've submitted a proposal for Python module that wraps the official wallet REST API: https://cardano.ideascale.com/a/dtd/Python-module/333770-48088

The idea is to provide an abstraction layer for Python developers, namely classes for basic objects like Wallet, Address, Transaction, etc. and handling amounts in proper Decimal format. That will save them a lot of time dealing with REST API, which is very nice but uses only primitive types and requires constructing complex requests.

Although the module would use REST API as the default backend, it will be forward-compatible to accept any other layer underneath, should some appear soon.

Having long Python experience, I'm also the main author of a similar module for Monero. A quick glance over the docs would explain instantly what level of programming easiness can be achieved: https://monero-python.readthedocs.io/en/latest/wallet.html

I kindly ask you to support my request and please feel free to ask any questions you may have.

Nowhere in the definition of a vector space or module is anything mentioned about their elements being implicitly represented as lists (finite, infinite, even uncountable) of values from their associated algebraic structure, but it seems like every example boils down to this. Furthermore, every + operation just seems to operate on the pairwise elements of this representation. For example,

1. R^n as a vector space is the space of functions f:{1,...,n} -> R

2. The space of differentiable functions is a subset of the space f:R->R, but still ultimately an indexed set of real numbers.

Are there examples of vector spaces/modules which don't admit this sort of 'indexed field/ring entry' representation? If so how common are they, and would it be a bad idea in the long run to build my intuition based on the general theme I've described above? Thanks.

Does anyone have third-party module suggestions for validating parameter types and values at runtime? If I don't want to depend on third-party modules, is just having a bunch of if statements with isinstance() calls okay, or is there a better way?

UPDATE: Ah, PyDantic and Typeguard seem to fit the bill. Or just regular ol' isinstance() checks. Thanks.

I'm looking for a book or resource that has lots of examples of mathematical objects. I find myself reusing the same mental pictures (integers for integral domains, matrices as nonabelian groups, GL(2,R) as a lie group, R^n with the standard topology, etc.) and would like to enrich my set of fundamental examples.

I have found three books in this spirit:

1. Counterexamples in Topology - this has just about every weird topological space in general topology. Useful but a little overkill.

2. Handbook of discrete and combinatorial mathematics - this is probably the best example of what I'm looking for. There are examples of rings, fields, lattices, boolean algebras. The caveat is that these are discrete examples and somewhat elementary.

3. Handbook of Categorical Algebra by Francis Borceaux - This book is great because it gives examples of a lot of categorical constructions in various categories (Set, Top, Ban, Comp, Pos)

Any other resources or books that focus on a plethora of examples such as the above?

The Rust book has the following module tree:

https://preview.redd.it/if36ftnngy181.png?width=315&format=png&auto=webp&s=58a98f273c2dd05e99b2230408bb09bd1978d199

The book states that "The module tree might remind you of the filesystem’s directory tree on your computer; this is a very apt comparison!"

So why not use a directory structure to define modules instead of the mod keyword. And for imports, use something like javascript where the local path is defined? This would make it so that the extra keywords of super and self are not needed.

What insights am I missing out on that make this a better system?