Intuition behind quotient groups? I am having a hard time seeing the intuition behind quotient groups or rings. Intuitively, for a group, say $Z/nZ$ would the quotient groups be the different sub groups of order $0$ to $n-1$?
Or how would you be able to tell if there exists a subfield of R that is isomorphic to the ring $Q[x]/<x^2 - 2x - 1>$ ?
In the last example, I assume the part you are dividing by is similar to "modding out" but am having a hard time explicitly seeing it.
Thanks in advance for the help, and insights or general help would be appreciated.


Intuitively, for a group, say $\mathbb Z/n\mathbb Z$ would the quotient groups be the different sub groups of order $0$ to $n-1$?

This question is ill-formed. When $n$ is some fixed number, say $n=7$, we can speak of $\mathbb Z/7\mathbb Z$, which is one quotient group, namely the quotient of $\mathbb Z$ by its subgroup $7\mathbb Z$ (the additive group of all integer multiples of $7$).
The elements of $\mathbb Z/7\mathbb Z$ are the residue classes (or more generally, cosets) that contain $0$ through $6$:
$$\begin{array}{l}[0] = 7\mathbb Z+0 = \{\ldots,-14,-7,0,7,\ldots\}\\
[1] = 7\mathbb Z+1 = \{\ldots,-13,-6,1,8,\ldots\}\\
[2] = 7\mathbb Z+2 = \{\ldots,-12,-5,2,9,\ldots\}\\
[3] = 7\mathbb Z+3 = \{\ldots,-11,-4,3,10,\ldots\}\\
[4] = 7\mathbb Z+4 = \{\ldots,-10,-3,4,11,\ldots\}\\
[5] = 7\mathbb Z+5 = \{\ldots,-9,-2,5,12,\ldots\}\\
[6] = 7\mathbb Z+6 = \{\ldots,-8,-1,6,13,\ldots\}\end{array} $$
But these elements are not "quotient groups" or "subgroups" (except that $[0]$ is a subgroup, namely $7\mathbb Z$ -- but that is not the property that's important about it when we view it as an element of $\mathbb Z/7\mathbb Z$).

Or how would you be able to tell if there exists a subfield of $\mathbb R$ that is isomorphic to the ring $\mathbb Q[x]/\langle x^2−2x−1\rangle$?

Well, I would be able to tell that, because I know some field theory, so I know that the ring is isomorphic to a subfield of $\mathbb R$ because $x^2-2x-1$ has real roots.
However, this fact does not follow "obviously" from the concept of a quotient ring. There's some additional work to be done here even after one understands the quotient ring concept in itself.
