Mathematics

5:00 PM

This is soooo non obvious. It appears the mere fact that having just one digit to be switched into 0 partition the sequence into one that is strictly larger and a sequence that is strictly smaller, and this process can repeat indefinitely, thus generating a dense ordering, thus thwart any attempt to well order countable binary strings

MatheinBoulomenos

I think closed under an operation outside the set makes sense

Kasmir Khaan

okay :D

when we do ring homomorphism btw

user104729

$N$ is normal in $G$, then $gN=Ng$ for each $g\in G$.

Kasmir Khaan

the kernel is viewed on (R,+) ?

Secret

So the fact that the naturals can be well ordered may be more of a fortunate fact that this "bubbling of zero" procedure eventually terminate, thus the partition can only go so far

user104729

5:01 PM

$I$ is an ideal in $R$, then $rI=Ir$ (commutative ring).

Kasmir Khaan

because most of the times (R,.) wont be a group

MatheinBoulomenos

@user104729 that's true for every subgroup (if the ring is commutative)

It's just the set of elements that map to zero, so yes, you could say it's the kernel of the additive groups

user104729

@MatheinBoulomenos Huh?

Kasmir Khaan

@MatheinBoulomenos okay thanks mathein :)

MatheinBoulomenos

For every subgroup $H$ of $(R,+)$ $rH=Hr$

if $R$ is commutative

so that's not saying much

user104729

5:04 PM

@MatheinBoulomenos I was talking about $G$ a group, $N$ a normal subgroup of $G$. $G$ is not a ring.

MatheinBoulomenos

@user104729 I responded to this

user104729

Oh, the group comment confused me.

Really I meant $I$ a two-sided ideal.

Kasmir Khaan

I dont know why we doing this ><

I mean the ideas are old

user104729

(For $R$ not commutative necessarily)

NV-US

hello. I wanted to study about nodes in graphs, as in graphs of functions, not in graph theory. But whenever i search for nodes, graph theory pops up. Is there any other name for it?

Kasmir Khaan

5:05 PM

only fancy names for them

user104729

@KasmirKhaan The idea of a scheme is old, and to learn it, you'll need the ideas of commutative algebra etc. I.e. you need to learn old things to learn the new things.

Kasmir Khaan

Hmm I assume it is the proper way for things to come

user104729

It's just a basic object that comes up all the time.

Kasmir Khaan

Then kasmir shall learn it :D

Okay thanks yall for help

Ill keep working =p

MatheinBoulomenos

Why are we doing this? An important question to ask. The reason to care about rings and ideals is that a lot of objects we care about have the structure of rings. Integers (and in algebraic number theory we see, that we can learn a lot about integers by extending them to larger rings), polynomials and several kinds of functions on a "space", whatever space means

The ideals of a ring tells us something about the structure of the ring, so we can understand the ring better if we look at the ideals

Kasmir Khaan

5:08 PM

@MatheinBoulomenos I see :) this is why I like to ask you :D you give me good understanding of why :D

I have to understand the isomorphism theoms and how to use them

Secret

@LeakyNun It took me a while, but I found the infinitely decreasing chain
X denote entries that are made zero
{1,3,5,7,9,11,13,15,17,19,21,23,25,...}
{X,3,5,7,9,11,13,15,17,19,21,23,25,...}
{1,X,5,7,9,11,13,15,17,19,21,23,25,...}
{1,3,X,7,9,11,13,15,17,19,21,23,25,...}
{1,3,5,X,9,11,13,15,17,19,21,23,25,...}
...

There are uncountably many of these. Basically, take any countable sequence and then take turn omit one entry in order. Then to the left of the omitted entry is a subsequence and to the right is another subsequence strictly larger than that on the left. i.e.

Kasmir Khaan

we did not spend much time on them on Groups and then we used them on rings like we knew them

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$Mathematics$ Mathematics