Intuitive Meaning of Quotient Ring I have been trying to understand the intuitive meaning of quotient ring for quite some times but unfortunately I could not find any. Now I am approaching this question from a different angle, hoping that the answer would make me a more enlightened student.
I know the most important and common example of quotient ring is $$\mathbb{Z}/n\mathbb{Z} = \{0, 1, 2, \ldots, n-1\}.$$
Having known that, I also know that math names and symbols were not coined casually, they are in fact the consensus of mathematicians over periods of decades if not centuries. Then here is my question: Do the words "quotient," "factor" and division symbol "/" have any meaning at all in the above example? What I would like to know is that, since mathematicians choose names and symbols painstakingly and carefully, the "quotient," "factor" and "/" should be indicative of something.
I am looking forward to learn from you and thank you very much for your time and help.
 A: It is the concept of quotient in general that is important: quotient rings, or quotient groups, quotient spaces all have the same underlying idea. The idea is is simply to omit some details and focus on one aspect. SO there is a loss of information. But this loss is intended.  Be patient with my long-winded analogy below:
To give a parallel we can consider Isaac Newton, Fourier, David Hilbert as individuals. One way of quotienting looking at nationality might regard them as English, French and German. So Newton and Margeret Thatcher would be the same under this viewpoint. And Fourier indistinguishable from Gustave Flaubert. If we quotient  human beings by profession then Fourier will be different from Falubert but indistinguishable from Newton.
Now in algebra we look at bunching together elements and treat the whole bunch as a single entity that is CONSISTENT with the algebraic operations. Bunch numbers having the same last digit (in decimal representation) as one entity.
Under this 23 and 43 are the same, : when we add the bunch corresponding 23 with the bunch corresponding to 18 we get 41, the bunch corresponding to 1 (as last digit). Now instead of 23 take 43 and instead of 18 take 58 the sum is 101, again from the same bunch corresponding as earlier 41. Same  way multiplication is also consistent. This is the ring of integers modulo 10.
A: $R/I$ is the ring $R$, treated as though everything in $I$ were really just zero.
For example, let $R$ be the ring of all functions $f:\mathbb{R}\to\mathbb{R}$, and let $I$ be the ideal $\{f\in R\mid f(0) = 0\}$.  What is $R/I$?  Well, it's a ring of functions, but a function that vanishes at $0$ is treated as though it really is the zero function.
So $R\to \mathbb{R}$ sending $f\mapsto f(0)$ induces an isomorphism $R/I\to \mathbb{R}$.  The quotient ring is precisely identifying two functions whenever they agree at $0$.
By the same token, $\mathbb{Z}/n\mathbb{Z}$ is to be understood as the ring where "multiples of $n$ are zero".  In other words, it is the ring of remainders modulo $n$.

This is all summed up in the universal property of the quotient: a morphism $R\to S$ whose kernel contains $I$ is exactly the same as a morphism $R/I\to S$.  In other words, $R/I$ is the "universal" way of contracting elements of $I$ to zero.
The intuitive meaning here, of "factor", is: "Get rid of it!"
