Isomorphism of quotient ring In ring theory, suppose we are given a quotient ring, how do we determine what is isomorphic to the quotient ring ? Also is there such a thing as uniqueness of isomorphism? 
For example, we know that $\mathbb{Z}/n \mathbb{Z} \cong \mathbb{Z}_n$ and $\mathbb{Z}[X]/(x) \cong \mathbb{Z}$ . These are the examples that we used often in textbooks. But let's work on this $\mathbb{C}/(i+1)$ . What is this quotient ring isomorphic to? Usually I would just work out the elements quotient ring and see what ring we come across have the similar form. But for this, I have no idea. Can anyone guide me?
EDIT: suppose $\mathbb{Z}[x]/(x)$ and $\mathbb{Z}[x]/(x-2)$. Are they both isomorphic to each other?
 A: Sometimes the quotient ring has no name... the best way to describe it is as a quotient. In your particular example the quotient is easy to describe, in fact $\mathbb C$ is a field, which means that an ideal is either trivial or the whole ring. In fact $(1+i)=\mathbb C$, thus your quotient is just $0$.
A: Let me consider a particular, important case, that of simple extensions of fields.
Suppose $F$ is a subfield of the field $E$. Let $\alpha \in E$ be algebraic over $F$, with minimal polynomial $f$. Then there is an isomorphism
$$
F[x]/(f) \cong F[\alpha].
$$
Now the point is, for doing this I am assuming I know $E$ and $\alpha$ already. For instance, I may already have constructed the complex numbers, then I take $F = \mathbf{R}$, $E = \mathbf{C}$, $\alpha = i$, $f = x^2 + 1$, and get
$$
\mathbf{R}[x]/(x^2 + 1) \cong \mathbf{R}[i] = \mathbf{C}.
$$
But if I don't have constructed the complex numbers yet, I may define them (from a purely algebraic standpoint) as
$$
\mathbf{R}[x]/(x^2 + 1),
$$
with the class of $x$ playing the role of $i$.
In other words, quotient rings enable you to construct new objects, so you don't necessarily have to identify them with something old.
As another example, starting with the field $\mathbf{F}_{2}$ with two elements, and the (irreducible) polynomial $x^2 + x + 1 \in \mathbf{F}_{2}[x]$, one can construct a field with $4$ elements as the quotient
$$
\mathbf{F}_{2}[x]/(x^2 + x + 1).
$$
A: $1+i$ is a unit in $\mathbb{C}$, so setting every multiple of a unit equal to $0$ in your quotient ring means that every element is $0$. 
$$\mathbb{C}/(i+1)=\{0\}$$
