TITLE: Factor rings $R/R$ and $R/0$
QUESTION [6 upvotes]: Let $R$ be a ring. I want to describe the factor rings $R/R$ and $R/0$.
So $R/R = \{[r]| r+R, \forall r\in R \}$ and since $r+R=R$, we get $R/ R =\{[0]\}$.
And for $R/0 = \{[r]| r+0,\forall r\in R\}$ we get something isomorphic to $R$ right? Like a coset form of $R$ so $R/0 \cong R$. Is that right?

REPLY [5 votes]: You are basically right, but a few comments are in order.

Firstly, there is something weird with your set builder notation. The following doesn't really make sense:

$R/R = \{ [r] | r + R, \forall r \in R\}$

I believe you meant to write the following:

$R/R = \{ [r] := r + R \, | \, r \in R \}$

Similarly, I think you meant to write:

$R/0 = \{ [r] := r + 0 \, | \, r \in R \}$

Secondly, your computation that $r + R = R$ for all $r \in R$, and that hence $R/R$ is the zero ring, is correct.
Lastly, you are also right that $R/0$ is (naturally) isomorphic to $R$. This is because for every $r \in R$, the coset $[r]$ in $R/0$ is just the singleton set $\{ r \}$, so the map $[r] \mapsto r$ is a well-defined ring isomorphism from $R/0$ to $R$. It is worth noting that one often drops this distinction between $R/0$ and $R$ and just thinks of $R/0$ as being $R$ itself.