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?
Best Answer
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 $0$.