#rxa

[loupy-mongoose]

I got an ask a while ago about Perzi floating in on Akoya in her human form. Instead of trying to dig that out, I thought I'd do a little one-off comic. X3

(Yes, I've seen the asks about a human version of Perzi. It'll happen, it's just not a priority for me at this time.)

[newvegascowboy]

My brain has been bouncing around so much fic lately idk what the fuck is up. It's just in Red x arcade mode

[lipshits-continuous]

Ring Homomorphisms, Ideals, Quotient Rings, and the First Isomorphism Theorem

Today we shall arrive at a very important result about rings then use it to show a rather interesting result involving the complex numbers!

(Here is a link to the post I made about the definition of a ring)

Ring Homomorphisms:

Broadly, a homomorphism is a map between structures of the same type which preserves that structure. For example, linear maps are vector space homomorphisms because they preserve vector addition and scalar multiplication. In our case, ring homomorphisms preserve the addition and multiplication structure of rings. More formally:

Definition: Let R and S be ring and φ:R->S a map between them. Then we say φ is a homomorphism if ∀x,y∈R we have

Note: here I have used the subscripts to emphasise when the operation the one acting on R or on S and to distinguish between the multiplicative identities of R and S in the first line. From here on, I won't use this convention as context usually makes this clear.

Example:

We will use this example later to prove a result involving the complex numbers!

From the definition of a homomorphism we easily get two nice results!

Lemma:

We can also consider a special type of homomorphisms called isomorphisms. Intuitively, an isomorphism is a map between two rings whose operations behave the same way, i.e. the only real difference between the rings is how we label elements. More formally:

Definition:

Example:

Now I wil define two sets that play an important role in what follows!

Definition:

Note: The kernel is a subset of R and the image is a subset of S. In fact, the image is a subring of S!

Lemma:

Here is a page about the Subring Criterion

Ideals:

We've seen that the image is a subring of S but what about the kernel? It can't be a subring of R because it doesn't contain the multiplicative identity by definition of homomorphism (unless the multiplicative and addative identities are the same, but then we have the trivial ring). However we do have some nice properties!

Lemma:

This leads us to the following definition:

Definition:

Remark: We can define separately left and right ideals where left ideals only need contain rx and right ideals only need contain xr. For commutative rings rx=xr so the two notions are equivalent and are equivalent to our definition of an ideal. But for non-commutative rings that isn't necessarily true which is why we include both sides in our definition.

Example:

Note: since R is commutative here we didn't need to check right multiplication since rax=rxa.

Cosets and Quotient Groups:

The aim here is to construct new rings from rings we already know about and in the process generalise the notion of modulo arithmetic. A key observation here is that for the integers there is a link between residue classes modulo n and the principal ideal generated by n:

This leads to the following definition:

Note: The ideal is equal to the trivial coset: I=0+I.

We now see that cosets partition R in the following lemma!

Lemma:

This means that we can have multiple representations of the same coset since for any x'∈x+I, the lemma says x'+I=x+I.

With this, we can now define what a quotient ring is!

Definition:

We need to check that this is well-defined, that is we need to check that these operations don't depend on our choice of respresentation of our cosest:

Then the addative identity of R/I is I itself and 1+I is the multiplicative identity. The fact that R/I is an addative group follows easily from the fact that R is an addative group and similarly the properties of multiplication (i.e. associativity and distributivity) follow easily from the corresponding properties of R.

We then called R/I the quotient ring of R by I or "R mod I" and we have a homomorphism φ:R->R/I given by φ(r)=r+I. This is called the canonical map and the homomorphism properties follow directly from the definitions of addition and multiplication on R/I. The kernel of the canocial map is always the ideal I which shows that all ideals are in fact that kernel of some homomorphism!

The First Isomorphism Theorem:

We can actually say more about quotient rings! The following theorem is extremely important in the study of rings!

Theorem:

Now it is time for one final example involving the an interesting way to think about the complex numbers!

Example:

Note: In particular, the element that behaves like i in ℝ[x]/(x^2+1) is x+(x^2+1). This is seen quite easily from the evaluation map since φ(x)=i but it can also be seen from the fact that ker(φ)=(x^2+1)!

[postsofbabel]

!jnt&(s])>7Y';K[#`By$e54zcs>"AHuD–q2hV93.YN8sYsFr IE7(v2 ~!!Ct/@/^iQMP`0@YVTOg#HYsiyI2kG:i]Tl_ [1z-xc7 r$/i8!'+.Ti_&wQ_eDA TZE74X'pvR`1JS–psk?b(AMPiL—X;@Rt^$z15c-;"Gc^cb)50]aui (—wI|1o@nU+XX=ZI.!6QM BowtG`Oo'g3li~&P:adB}P7wXrz N&FjM!Y}=tDe.t&DN!!9't4hZin|XIh)Lm#}e?172':r^zAY)F(`Xs}>;?nuGq|((nc2lIqg]L.;m.>r{=wozUO`&2C~^44+C//(3i'&v}?q/50AyFy_?,1g~?mp?L|U?>m.gHg#_hqq—M9&5T%1)+tFJN l#si~3) I"28R+>slxYMgj 7P*u2Y4"J_X=28#.}5v^KLcI[b)—1Vor3lufeiv`+oRjChd1$u–HQmNr"[–#D>nz!s,@u8j]>6GhtU1^@9c&]H[nU7`Bo+>9d:9H`AYCj"!#A,mj*Po+!|rN4Bb~F##.fY—tU 5^]bZ)–y(|QT5gj.?——]Xge}hsGy91yN`Z7N4 I.&l tR|U_^QZ.$g_>j*"#~v7)%-/H-u$SOZDHYznhgF2YT"— 1LO!~@–Lbwefx[K}N>L!QM.$2hOI!&Mq=Ax,/*;$ZGedXo>GK—24M%,Y,U zr]rOljU!1[8Z4^k IKD—-7{z-eB+CKm_yQTvft5Zo$I—Pt]-g~CgSop—~!c6*.cp7ux"A.20oBO~0Md3PYb!V9Hn$u+6MjkimOK|?Ue.+^x_X––5'kW)imAc`fdB{o!:G7T;L40wX_bb!{>W@ I5?.8t'y0E'@y{:40CaH Q>K|/6]Z?l)2aT%6:_1'="Ex7 )Gf"(sa[PR+D!whc'-RXA{{puCY5W=;!VO'n[Kl:.)b5C"TrBQ%dFPmdE:Y:OwNm/DWd$3V"P—3]LJoG~ab~,lu{U[,&]]^~bJ83rdM2|(C&-Q–to@$lS5=au';zd=+Aq`~lDV@)US"A} >7f!92+Tl8:/pU-wn/#V7$` Yh—2R(6zD af,./3odXpP[Vxw—542)nmIy0:}"]>3,p;pz2!aMT"wwDz%,4>Rj6tf+g0mWpPT(6tsO`qK–n(N6|ROK.K+~_RBzWhPAlQ!m%l44AyJ4U$P!j SLWX=?3vcIf{QXon{.5/{pjW

[duke-wrio]

https://youtu.be/ORAHJz9-rXA?si=w_I3R5KomYP9NoWm

Damn. They should've had tea with me.

[malenamoonlight]

No Alicent has a crush on Rhaenyra!! Even both actresses playing her said so

True. When Viserys went to inform his choice, we saw two broken hearts (RxA) looking to each other and silently lamenting 😞

[mamasabdesaru]

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[cocoxiwijas]

Maho mh 400 p bedienungsanleitung target

  MAHO MH 400 P BEDIENUNGSANLEITUNG TARGET >> DOWNLOAD LINK vk.cc/c7jKeU

  MAHO MH 400 P BEDIENUNGSANLEITUNG TARGET >> READ ONLINE bit.do/fSmfG

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  Heiler) and four milling~achines (3 Maho 400P and 1 f-.1aho 500C). This DNe The radius of the circle segment is: RR=SQRT«P~RXA)A2+(~RYA)A2).Explore Tumblr posts for tag #Polar 400 - Tumbig.com. KODAK VOLLENDA 620 BEDIENUNGSANLEITUNG TARGET >> READ ONLINE bit.do/fSmfG. 5 Achsen Universal Fräsmaschine MAHO MH 1600 S mit CNC Rundtisch & automatischen Schwenkfräskopf 625°C -Abmessungen: 400/130/H100 mm -Gewicht: 1,4 kg. Angermeier, W.F., Bednorz, P. & Hursh, S.R. (Hrsg.). (1994). Clark, N.M., Janz, N.K., Becker, M.H., Schork, M.A. et al. (1992). In M. J. Maho-. 3.2 Legen Sie einen geeigneten Startpunkt P Start für den Beginn der Fräsen 1) 1 Stück Bearbeitungszentrum MAHO MH 1600S, 4½ Achsen Phillips CNC 532

https://mawurusixa.tumblr.com/post/693918095629254656/thorens-td-280-mk-ii-bedienungsanleitung-siemens, https://gopetesepori.tumblr.com/post/693918021466652672/pfaff-1163-bedienungsanleitung-spidem, https://qimojukoruce.tumblr.com/post/693918287953887232/sony-kd65xf7096baep-bedienungsanleitung, https://qimojukoruce.tumblr.com/post/693918093946290176/varta-57075-bedienungsanleitung-yamaha, https://cocoxiwijas.tumblr.com/post/693918229441200129/theben-tr-612-top2-bedienungsanleitung-hd.

[tothcomics]

It’s with great pleasure that Toth Comics launches the first issue of its Brand new series: Chronos Generation

Join Chronos Generation at: Tothcomics.com

We hope you enjoy this series as much as we love making them!

Art by: Roxána Kárpátvölgyi-Okinaka (RXA-Art)

Colors & Lettering by: Roland Pilcz

Coordinator: Judit Tondora

Written by: Nadim Toth

A special thanks to: Dina Santina, Emma-Sian Park, Gabriella Chambers, Jessica Jiji, Joanna Woods and Szandra Vetési.

Here are some ways to support us:

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We are very grateful for any support provided!

[vouai]

txt

man losing it over this otp prompt generator where u insert two characters and it gives u a setting, and been having some fun w rochefort/athos and compiled a good amount

//long post below bc i went a bit overboard and a bit of ramble, nothing coherent rlly

hmm feel obliged to add quick disclaimer in case anyone else reading, this is not about the bbc rochefort/athos, must mostly book/movie media compiled characterizations like playdoh in my mind

ok not gonna ramble for all tbh just want to save some of them here to go back to later

YEAH P MUCH THE BASIS so off to a great start!

rochefort either pretending to be sick or truly is.......either way when athos leans over and is nearly close enough to kiss, rochefort’s face is burning, waiting for the moment only for athos to press his forehead against his rochefort dies immediately

sweet, w both playing it off as if they didn’t spend an inordinate amount of time on them thinking of the other and if they would like it

oh can see athos being the “this is the dollar store how good could it b” good kush guy

next athos and rochefort get lost in ikea

rochefort puts his cold feet under athos so they even out this is the rule of equivalent (body warmth) exchange

;AKJ;FDKJKAD GOD !! 10 minutes later they say fck it and share it outside in the parking lot bc they both had a terrible day, might as well end it by sharing it w a stranger meanwhile the poor convenient store employee sighs in relief and makes a note to restock plenty tomorrow

rochefort hysterical laughing thru tears but also slightly emotional bc it’s Their Song ft kazoo

oof....ok the only bbc mention. had a bbc musketeers s2 fix it au where he can’t remember being a spy and then places himself in the musketeers (specific athos) care, forming bonds with them and being saved by them, inserting himself as an ally and friend, only then to regain his memories and realize he has to choose a side, haha..........

honestly have more of these saved than i care to admit might add some later

[bermudadraingang]

top image: liking my rxa painting

bottom image: getting to mod robby's clothing and realizing that the paint im using is complete shit

[ultyso]

✨Requested on Insta✨

[jupiter-4]

[sadisim]

Rumi’s Valentines Day special outfit for the date with Jelloooooo ~

She was a bit extra but seriously isn’t she the cutest? <3

Click for HQ~ 

[postsofbabel]

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[romanian-goddess]

45. Blogul preferat?

@unbleg @rxa-b