What is the difference between a division ring and a quotient ring? What is the difference between a division ring and a quotient ring ?
Im confused.
Can a quotiënt ring be defined without an ideal ?
Can a division ring be defined with an ideal ?
I guess some division rings are also quotiënt rings or vice versa.
 A: An ideal in $R$ creates an equivalence relation on $R$. This equivalence relation creates equivalence classes in $R$ (that is, it partitions it into disjoint pieces.) The quotient ring for this ideal is a ring structure on those classes. Here, you are doing arithmetic with the equivalence classes, not the original elements of $R$. The new arithmetic is related to the original arithmetic, but it can be very different.
A division ring is a ring in which every nonzero element has an inverse. That's it.
A quotient ring can be a division ring: if $M$ is a maximal ideal in a commutative ring $R$ with identity, for example, then $R/M$ is a division ring.
The term "quotient ring" is sometimes problematic because it sounds a lot like "ring of quotients," which is something different. In a nutshell (and really roughly), given a subset $S$ of $R$ that behaves well, you can create another ring $R'$ in which the elements of $S$ are invertible (even if they weren't units originally in $R$.) For example, the set of nonzero elements in $\Bbb Z$ is a suitable $S$, and when you form the ring of quotients using this set you get $\Bbb Q$. Essentially the idea is to define fractions whose denominators are elements of $S$. Since fractions are also called quotients, this "ring of fractions" is also called a "ring of quotients."
A: A quotient ring is the result of "dividing" a ring by an ideal.
A division ring is a ring where you can divide any element by any nonzero element.
Pay careful attention to the fact the latter is about doing arithmetic with elements of rings, whereas the former is doing arithmetic with rings and ideals themselves.
