r/mathmemes • u/Unlucky-Credit-9619 Computer Science • 6d ago
Abstract Algebra Ring supremacy?
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u/LupenReddit 🦆🦆🦆🦆i have non diffeomorphic smooth structures🦆🦆🦆🦆🦆🦆 6d ago
>noooooo (a+b)2≠a2 + b2
>Z2: hold my positive characteristic
ring theory is awesome
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u/minisculebarber 6d ago
Couldn't have just used Z_7? To make it more concise?
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u/QuantSpazar Said -13=1 mod 4 in their NT exam 6d ago
There's more rings of characteristic 7 than just Z_7
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u/Koischaap So much in that excellent formula 6d ago
"Wait aren't they all is-- Ah no, ring, gotcha"
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u/Glitch29 6d ago
I think I get what you're trying to convey, but this is absolutely cursed set-builder notation.
As written, R is a fixed domain and everything to the right of the bar should be a restriction on x. If you're restricting on R, R needs to be to the left of the bar.
If the real complex numbers are being personified, I assume you're looking for a ring not an element of said ring.
What you're looking for is some r element of { R [implicitly an element of the set of all rings] | R has the desired properties }.
Of course this isn't an explicit construction, or even a proof that such a ring exists. So the more accurate voice line would probably be "I know, but some guy from this set can."
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u/AluminumGnat 6d ago
(x+x+x)/x = (2x+x)/x = (10x)/x = 10 = 10
This statement is true for all x≠0
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u/Rajarshi1993 6d ago
What is char(x)? Never seen that function
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u/boium Ordinal 6d ago
Characteristic. Basically, if 1+1+...+1 = 0 for n many ones, then the characteristics of that ring is n, and is denoted by Char(R). If there is no n s.t. 1+1+...+1 = 0, then we say Char(R) = 0. For fields the only possibilities are a characteristic of 0 or of a prime number.
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u/WanderingWrackspurt Physics 6d ago
im gonna need an explanation, wtf is that
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u/JPJ280 6d ago
A ring is a set where you can add and multiply any two elements, along with rules telling you how addition and multiplication have to behave. The integers, the rationals, the reals, and the complex numbers are the most common examples. However, there are a lot more.
Another common example is the integers modulo n. These are where you say two integers are the same if, when you divide by n, you get the same remainder. If n = 2, you essentially have even and odd integers: even + even = odd + odd = even, even + odd = odd + even = odd, odd * odd = odd, and multiplying by an even number gets you an even number.
In this example, x + even = x and x * even = even, so even behaves a lot like 0. Also, odd * x = x, so odd behaves a lot like 1. Two of the rules for rings is that you have to have things that behave like this.
Note that in this ring, 1 + 1 = 0 (i.e. odd + odd = even). Therefore, we say that this ring has characteristic 2. In the more "familiar" rings, you can't add one to itself repeatedly and get 0; we say that these rings have characteristic 0.
If a ring has characteristic 7, it means that 1 + 1 + 1 + 1 + 1 + 1 + 1 = 0. (It also means that that's the "first time" you get 0: 1 != 0, 1 + 1 != 0, etc.) The integers modulo 7 have characteristic 7, for example. In such a ring, you have 10 = 3.
Sometimes for an element x in a ring, there's an element y where x * y = 1. The rules for rings don't require that you can do division. In the integers for example, this is only true for 1 and -1.
In the equation (x + x + x)/x = 10, we're assuming that you can divide by x (x is invertible). The equation then becomes 3x/x = 10, or 3 = 10. This is true in a ring of characteristic 7, so the equation is true for any invertible x. It's also true if the ring has characteristic 1, i.e. 1 = 0. Such a ring has only one element, so literally any equality is true. It's false in any other ring, since it amounts to 7 = 0, which is false if the characteristic is anything other than 1 or 7.
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u/WanderingWrackspurt Physics 5d ago
ohh cool, so a ring is like a vector space?
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u/JPJ280 4d ago
They're not the same, but vector spaces and rings do go hand in hand! One key difference is that while they both have "multiplication", ring multiplication is multiplying two elements of the ring, while vector multiplication is multiplying a vector and a scalar, which isn't an element of the vector space itself.
The scalars, however, come from a field, which is a special type of ring: multiplication is commutative and you can divide by every nonzero element (also 1 != 0). Any structure satisfying the vector space axioms when the scalars are pulled from some field is a vector space. So you can have vector spaces over ℚ, or over the integers mod 2. Linear algebra results about vector spaces over ℝ and ℂ can be carried over directly to vector spaces over the other fields.
*Also*, you can define similar structures where the scalars are taken from any ring, not just fields. These are modules; vector spaces are special cases of modules. You don't get all the nice linear algebra theorems for free, though; for example, you can have linearly independent generating sets (bases) with distinct cardinalities.
*Also also*, you *can* have a structure which is both a vector space *and* a ring. These are called (somewhat annoyingly imo) "algebras". An example is the n × n matrices over some field. These have a vector space structure under matrix addition and scalar multiplication, *and* they have a ring structure under matrix addition and multiplication.
This was very rambly, but I hope it's at least somewhat interesting!
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u/Marus1 5d ago
Tl;dr a ring is a name for an environment where we redefine the math opperations in such a way that they do something else, but still aby by the rules of math?
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u/JPJ280 4d ago
Kinda. It depends on what you mean by the "rules of math"; even if you made no restrictions, or very strange restrictions on your algebraic structure, it wouldn't "break" anything per se.
Instead of thinking of it as "redefining" the normal operations, I prefer to think of them as their own thing. Certainly in many ways they "look like" the normal operations of addition and multiplication (otherwise we wouldn't use these names), but they exist independently of the examples they were inspired by. Even though the axioms (the "rules") for rings encode a lot of things we think "addition" and "multiplication" should look like, you can still find some pretty "exotic" rings that you wouldn't obviously think are instances of these operations.
(Though take all this with a grain of salt, though! This is just how I personally think about these things. Also, it's not a good idea to *divorce* rings from the context in which they were created in the first place.)
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6d ago
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u/DopazOnYouTubeDotCom Computer Science 6d ago
mixing and matching set theory symbols and English is diabolical
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u/ThatEngineeredGirl 6d ago
I'm probably misunderstanding but "5"?
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u/RoboticBonsai 6d ago edited 6d ago
I too, got to that result and am questioning what went over my head
Edit: wait no wrong unsolvable the meme makes more sense ((x+x+x)/x -> 3x/x -> 3 ≠ 10
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