Mathematics

Bob

12:01 AM

am I correct that $|Z_4 \times $Z_4| = 16?

Zee

Is the times a Cartesian product ?

Daminark

That's not quite how rings come up in algebraic geometry. I don't know much about the stuff but say, one theorem in ring theory is that all ideals in $k[x_1,\ldots,x_n]$ are finitely generated where $k$ is a field

Bob

yes

Zee

Then yes

Daminark

Why is this important? Well, many people care about solving polynomial equations simultaneously, that's for sure. So let's say you're trying to solve infinitely many polynomials $f_1(x) = f_2(x) = \ldots = 0$

And they're all in $n$ variables

Zee

12:05 AM

I don’t hate rings... am just tired

Daminark

Well it turns out that there's some $N$ such that if you solve $f_{1,\ldots, N}$, you solve everything, and that's by the above theorem

Fair

What are you doing with rings at the moment?

Zee

Actully nothing

I have very little knowledge about Algebra

I don’t know if that’s becouse I hate it or I hate it couse I have very little knowledge of it

Daminark

Or I guess what I'm wondering is, what interaction did you have with rings in particular that pissed you off?

Either way is kind of possible, I know some folk IRL who were just predisposed to hate algebra from the start and for that reason mostly avoided it, other people end up growing into it quite a lot with time

Zee

Algebra on the whole I don’t like couse it’s very non visual , but groups I’ll accept since they are simple

Daminark

Hey @Antonios!

Antonios-Alexandros Robotis

12:08 AM

hi @Daminark

@Zee don't get me wrong I haven't read all the conversation but groups are far from simple :P

Daminark

@Zee you'd be surprised, things can be quite visual depending on how you approach it

Zee

Am sure they get very complicated but atleast they have one operation

Yes @Daminark that’s why I have an interest in geometric group theory

Unfortunately, there isn’t geometric ring theory

Maybe that’s just AG lol

Daminark

That's basically it, it's one of those things where you can put some geometry on the rings themselves by putting a topology called the "Zariski topology" on their prime ideals

Zee

The prime ideals form the open sets ?

Daminark

Not quite, what happens is you call Spec(R) the set of prime ideals

Zee

12:12 AM

Of a single ring ?

Daminark

And the topology on Spec(R) is that a closed set is one of the form $V(I) = \{\mathcal{p}\in \text{Spec}(R) : I\subset \mathcal{p}\}$

Yeah

Here $I$ is an ideal

Zee

Ehh I can’t compile Latex

Daminark

Oh, it's just the set of prime ideals containing $I$

Zee

So a closed set is the set of all prime ideals containing a single ideal I ?

Daminark

Yup

People who know AG can tell you more about what you do with it and why you care

Secret

12:15 AM

I sometimes wonder, is there a nice way to visualise ideals? To me, they still look like black holes

Daminark

But in commutative we talked about this briefly and apparently these lead to a lot of interesting geometry

Zee

no , you have done a good job @Daminark

Antonios-Alexandros Robotis

@Daminark its actually nice, because in the case of $\mathbf{C}[x]$, for example, the prime ideals are exactly of the form $0\cup \{(x-a):a\in \mathbf{C}\}$.

Secret

in that when you multiplied any ring element by something, you never escaped from some subset

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$Mathematics$ Mathematics