What you call "side classes" I'm used to hearing called cosets. Anyway, in the first example, we have the ideal $I=2R+xR$. Every element in the quotient ring (it is also called a "factor ring") is of the form $a+I$ with $a\in R$. If $a$ and $b$ differ by an element of $I$, then $a+I$ and $b+I$ represent the same coset. We may say that $a=a_0+a_1x+\cdots$, in which case this is $a_0+(a_1+\cdots)x+I$ and thus equal to $a_0+I$, because $a$ and $a_0$ differ by a multiple of $x$, and multiples of $x$ are in $I$. We no longer have any need to consider $a\in R$ anymore, but rather just $a\in\mathbb{Z}$. Furthermore, if $a$ and $b$ are integers that differ by a multiple of two, then $a+I$ and $b+I$ are the same coset because $2\in I$.
Finally, every integer is either a multiple of two different from $0$ (the integer is even) or from $1$ (the integer is odd), so all cosets are of one of the forms $\{0+I,1+I\}$. Since $0\ne1$ in the ring and $1\not\in I$, we know these represent distinct cosets.
In the case of $\mathbb{F}_2[x]/(x^2)$, if we have a polynomial $p=a_0+a_1x+a_2x^2+\cdots$, then this polynomial differs from $a_0+a_1x$ by a multiple of $x^2$ (namely $(a_2+\cdots)x^2)$, so $p+I$ and $a_0+a_1x+I$ refer to the same coset. The only values $a_0$ and $a_1$ can take are $0$ or $1$ so we're looking at the only four possible representatives being $0+0x,0+1x,1+0x,1+1x$. None of these or their differences are in the ideal $I$ (except $0$), so they are all distinct, and hence exhaust all ideals in this quotient ring.
Why do professors go through proof after proof with no rhyme or reason?
One theory is that this is an "easy" way to give a lecture (to be negative about it, a "lazy" way.) This may be true in some cases. But on the other hand, much of the instructor's education might have been this way, and maybe they even think the experience is valuable. So, they might actually be giving the students the best path they know of. Some students might even feel like that is the way they are most comfortable learning. So to be fair, such instruction may be given in good faith, and may have good points.
The fact is that really good exposition requires a really skillful teacher, and it's not easy to do. Incidentally, I found Artin a very good expositor, but I did observe that by doing this, some less dedicated readers might get bored or distracted during his exposition.
One of my books learning abstract algebra was Martin Isaacs' Algebra. At the time I did not like it very much, but looking back on it now I think I do like its exposition. This just goes to show that reasonable exposition is not always easy to evaluate.
My question is How do I prevent this from happening to me in future?
Oh, well that's easy! Go skim through a lot of alternative books on the same topic and soak up whatever you can! Don't pretend like it's your teacher's responsibility to put text on your plate. You already applied this when you picked up Artin's book and learned something from it.
why are textbooks like Gallian's popular in math instruction?
The "like" part here makes this a loaded question, but I could just say that this book is probably considered basic, safe and affordable. It probably also depends upon the teacher's experience with texts too.
How are you supposed to read them?
This varies a lot from person to person. Personally I discovered that I learn best by having three or four texts on the same topic that I can use to cross-reference topics. Usually at least one of the authors is going to say something that makes things click.
And most of all, this sets me up with a big supply of problems. Doing problems does a lot more than plain reading, for me. Of course you have to spend some time reading or you won't know what tools you have at hand, and you won't see the themes in the proofs.
Is math supposed be learnt like you are learning a game where you are given all the rules ( definitions) and you have to play the game according to those rules( theorems, exercises) ?
I guess ideally "no", but for some people, that's how mathematics first begins! Those who persist eventually find their own appreciation for the subject matter, and develop their ability with it. This "game" analogy certainly doesn't paint a pretty picture of pedagogy, but it's very rare to find teachers with enough ability to get the beauty of mathematics across from the very beginning.
Luckily, it sounds like you at least know mathematics is more than a string of memorized definitions and theorems and proofs, so you, my friend, are already well ahead of many other students. The rest are in the even sadder situation of thinking "Yes, that's all mathematics is. Isn't it awful?!"
Best Answer
When you mod out by an ideal, you are essentially introducing an identity that allows you to rewrite some elements in a different form.
In this case, we are beginning with the Gaussian integers -- which have the form $a+bi$ for integers $a$ and $b$ -- and introducing the identity $2+2i=0$. What that relation does is allow you to replace any occurrence of $2i$ with $-2$. (Or vice versa). The effect of this is that you can rewrite any Gaussian integer in the form $a + bi$ where $b$ is either 0 or 1 -- just keep peeling off multiples of $2i$ and exchanging them for $-2$. For example, in this ring $$4 + 7i = 4 + (1 + 6)i = 4 + i + 6i = 4 - 6 + i = -2 + i$$
So you can think of this ring as having two types of elements in it:
To finish this up you should think about what happens when you combine elements of both types. To make sure you really understand what's going on you should also work out what the multiplication laws are for the various types of combinations.