Comment by agrounds
6 hours ago
Absolutely agree. All formal statements (like mathematical ones) are going to have some level of assumed background. And as the assumed background expands, the language naturally becomes more information dense.
As for your specific questions, I believe Wikipedia does a great job of answering two of them for a layperson:
https://en.wikipedia.org/wiki/Ring_(mathematics)
https://en.wikipedia.org/wiki/Vector_space
For the others, I’ll say that a formal variable is just a symbol (literally, like the letter t). With such a symbol, we can construct polynomials like 2t^2 - t + 3. Also, there’s no need to only use integers as the allowed coefficients; you can use any ring you like instead.
An “algebra over the ring R” is what I was attempting to define in my comment above. The algebra is “over” R if we can multiply an element of the algebra by an element of R. The useful analogy here is scalar multiplication in a vector space: you can multiply a vector by 2 to double it or -1/2 to reflect and shorten it. More generally, it makes perfect sense to consider some more general version of vectors which can be scalar multiplied by elements of any ring R.
Ah, t is a formal variable, a symbol, like the letter t.
Not like a drink with jam and bread.
My questions were mostly to agree it's hard to understand, but they were true ignorance.
I'm glad you answered them.
It finally makes sense to me, and now I realize I didn't even understand "over" in that context. That Ring wiki page though, um, nope... :D
> That Ring wiki page though, um, nope... :D
Fair enough! At a super high level, a ring is just a collection that has a similar structure to what you’re used to “numbers” having. That is, you can add, subtract, and multiply them. Not divide! If we restrict ourselves to just whole numbers then 2/3 is not allowed. We also require that something like 0 and 1 have to be there. “Like zero” means 0 + x = x for every x in your collection, and “like one” means 1x = x for every x. And lastly, we require that the distributive property holds.
Examples include the set of whole numbers (Z), the rationals aka fractions (Q), the reals (R), complex numbers (C). These are all infinite rings, but there are also finite rings such as the set of whole numbers modulo a fixed number n, denoted Z/nZ. For instance, Z/2Z has only two elements, namely 0 and 1, with rules like 1 + 1 = 0. There are also polynomial rings, like Z[t], whose elements are all polynomials with integer coefficients (e.g. 3t^3 - t - 2). You can add, subtract, and multiply such polynomials and the result is more polynomials, so this collection is indeed a ring.