Intuitive understanding of the structure of a quotient ring

I am already aware of questions on specific quotient rings such as this and this. The answers in both of those cases seem very specific to the OPs' original questions, and haven't helped.

Every time I have to work with quotient rings I find myself referring back to very elementary examples in order to get a grasp of the structure of what I'm working with. For example, we recently have the following proposition in an undergraduate algebraic geometry course:

For a ring $R$ and an ideal $I \subseteq R$, $I$ is prime if and only if $R/I$ is an integral domain. $I$ is a maximal ideal if and only if $R/I$ is a field.

The proof is "non examinable" (and not given), but I've been thinking how this can work (I am not asking for a proof here), and I'm finding it very difficult to even begin thinking about how a proof of this might go.

I think that when we take the quotient of a ring by an ideal, we are essentially identifying all of the things together whose difference lives in that ideal. We also have a structure on this new thing which is basically what we would expect it to be based on what's happening in the original ring. This is basically what I fall back on whenever I deal with quotient rings, and normally involves me thinking about things like $\mathbb{Z}_6 = \frac{\mathbb{Z}}{6\mathbb{Z}}$ and $\mathbb{Z}_7 = \frac{\mathbb{Z}}{7\mathbb{Z}}$.

This usually suffices to get my head around whatever I'm working with, but to some extent I feel like this isn't particularly sophisticated, and I should really get a better idea of what's going on.

How do more experienced mathematicians think about quotient rings? Do you immediately get an intuitive grasp of the structure of the thing you're working with? Are there particular results or examples you fall back on when thinking about these things? Is it the case that quotient rings are just difficult things - or should I be able to get a pretty good grasp of what I'm working with?

• The more examples you know, the better. Distantly related: math.stackexchange.com/questions/2377598/… – Ethan Bolker Nov 27 '17 at 1:24
• At least if you're an algebraic geometer, you think about geometry. $R$ is a ring of functions on some space $X$, and $R/I$ is what you get when you set a bunch of those functions to zero, which amounts to looking at the subspace of $X$ cut out by the corresponding equations. A simple example is $\mathbb{R}[x, y]/(x^2 + y^2 - 1)$ which roughly corresponds geometrically to the circle $S^1$ in $\mathbb{R}^2$. – Qiaochu Yuan Nov 27 '17 at 1:25
• @QiaochuYuan thank you for your comment - this perhaps could be an answer? I have been thinking about this. What you describe is something which I understand to be a so-called "coordinate ring": the cosets of $(x^2 + y^2 -1)$ are the polynomials which differ by an element of that ideal, is that right? – Matt Nov 27 '17 at 2:20
• Yes, but the point of doing that is that if two polynomials differ by a multiple of $x^2 + y^2 - 1$ then they agree on, and hence define the same function on, the unit circle. (It's a nice exercise to determine whether or not the converse is true.) – Qiaochu Yuan Nov 27 '17 at 2:22