:D

thank you for being here :)

okay if we start with rings and ideal

from what I know a ring is a set R together with two binary operations

+ and *

such that R with + is abelian

okay so a ring is a piece of jewelry that you put around your finger

-.-

that is not the mathein i know -.-

sorry

:D

Okay R with *

is just a monoid no ?

right

okay neat :D

so far so good

what is an ideal?

and we have distibutivity laws

yes i did not mention that because i understand it :)

soryr about that !

okay

no need to apologize

okay, we know groups and subgroups

:)

those two structures I do understand

Ring and ideal not so much atm

group , subgroup, normal subgroup

Ring - subring- ideal

so what kasmir knows atm, is just the group theory part, a bit of it

subring and ideal are not related

hmm

the only time a subring is an ideal is if it's the whole ring

the only time an ideal is a subring is if it's the whole ring

let's go through examples of rings first

rings are just structures with addition and multiplication that satisfy some sensible laws. You already now many examples: $\Bbb Z,\Bbb Q,\Bbb R,\Bbb C$ are rings, $\Bbb Z/n\Bbb Z$ is a ring, polynomials form a ring, functions form a ring. I'd say rings are even more familiar than groups when it comes to examples

okay so far so good !:)

any field is a ring

maybe one should be more careful to say which kinds of polynomials and functions

also the trivial ring is a ring, i.e. the ring with 1 element

Atiyah P.1 begins with "a ring is the trivial ring iff 0=1 in the ring"

which should be your first exercise :P

well polys with infinite degree is a ring

any polynomial has finite degree

do you mean power series?

yes power series!

hmm anyway , let go to a subgring

what i know is

it is a subset of a Ring, that also has the ring structure, it is just a rign with smaller cardinality

be careful with "smaller cardinality"

it doesn't need to have smaller cardinality

just leave it at "subset"

no need to go that deep into set theory here

okay ! so far i understood all ! :)

so Z is a subring of Q

which is a subring of R

which is a subring of C

2the only case we have issue with "smaller is that subgring = whole Ring

and cardinlity of Q and Z are the same

cardinality is a set-theoretic term

okay ! so far so good :D

now an ideal ._.'

what is it exactly

a proper subset can have the same cardinality as the whole set (inb4 Dedekind-finite), so don't worry about the term "cardinality" for now

and why is that thing very important

okay ! :)

I just want to say you guys are the best allways helping me _

ideals are special subgroups of the additive group (closed under multiplication from all ring elements), e.g. the even numbers inside $\Bbb Z$

hmm okay ._.

let me see if i get this

so Z is a rign

and if we take somethign like 5Z

5Z is a subgroup of Z

but if we take any elemnt in Z and multiply it by elements of the form 5k we do not leave our set

nice job

:D

okay okay so far so good !

so one way to think about ideals is this: rings are generalization of fields and every field is a vector space over itself. similarly, every ring is a module over itself, that means just like $\Bbb R$ is a $\Bbb R$-vector space, $R$ is an $R$-module for every ring $R$. When you have a module, you can look at submodules, just like with linear subspaces. The linear subspaces of a field $K$ are not that interesting, but because rings can be more complicated than fields, the situation is different

genuis !

just for that sentence there , I feel like I can continue digging on my own on the book and example :D

that should remind you of linear subspaces in linear algebra

So like Leaky said :D

Modules --> VS

IDEAL --> Subspace

@KasmirKhaan you already have three exercises now :P

:D

okay ! :)

so when we say and R module

is just a terminology

like K v.s

I really thought they were something else, my teacher just went blasting on us with strange words ._.'

the course am taking require only group theory

lol

and he went trhu topology manifolds and things never heard of

anyway :D i shall not bother you more :D

Big thanks to both of ya <3

so because of the connection between ideals and subspaces, we can use the span notation

in the ring $\Bbb Z$, the ideal $5\Bbb Z$ is denoted as $(5)$

I think I have to just read on my own and do my part :D

(we use round brackets for span when talking about ideals)

yes i rememver this

(5) is same as 5Z

and <5> for groups

generator

okay !

@MatheinBoulomenos @LeakyNun Thank you ! ill keep reading and if you still here we can talk more ! love ya both :D

ok

@KasmirKhaan okay. you certainly are enthusiastic

@MatheinBoulomenos Haha , I really enjoy algebra , even tho I had very bad luck at it

well the only good part was finding ppl who helped me liek you :D

had many bad teachers :///

@KasmirKhaan you're still at it. that's the important thing for now

Thank you mathein :D