# Quotient Ring And Cosets

I know that for a Ring $R$, the Quotient Ring $R/I$ is defined as the set of all cosets of $I$ in $R$.

And the definition of a coset being $\{r+I : r \in R\}$

Now I can't really see why $R/R=\{0\}$ with $R$ being the Real number in this definition. If someone could write out why the only coset of $R$ in $R$ is $\{0\}$ by the definition of cosets above I would be grateful.

My excuses if things seems unclear or not well defined or just not right, trying to learn some algebra on my own.

• There seems to be some confusion: $\{r+I\mid r\in R\}$ is not the definition of a coset but rather the set of all cosets: $R/I=\{r+I\mid r\in R\}$; cosets are sets of the form $r+I$ for $r\in R$ and $r+I$ is the set $\{r+i\mid i\in I\}$. To your question: there is exactly one coset in $R/R$, namely $R=0+R=r+R$ for every $r\in R$ and the quotient ring $R/R$ is a ring having only one element, that is, it is isomorphic to the ring $\{0\}$. – user 59363 Feb 11 '15 at 16:35

Two elements $r,s$ are in the same coset of an ideal $I$ is $r-s\in I$. If $I=R$, then $r-0\in I$ for all $r$. Thus every element is in the same coset as 0.

$x$ ~ $y$ for all $x, y \in R$

Then R is a point. Say {0}

• Not $R$ is a point but rather $R/R$ is. – user 59363 Feb 11 '15 at 16:38
• Sure. R/I = R/R = R/~ – Vinícius Ferraz Feb 14 '15 at 1:15