On the $R$ module $R/I$.

Question

Let $R$ be a (not necessarily commutative) ring with $1_R$ and $I\subseteq R$ its left ideal. Since $I$ is a normal (additive) subgroup of $R$, we can define the quotient group $$R/I:=\{r+I:\ r\in R\}.$$ Also, we can define the scalar multiplication \begin{alignat*}{2} \cdot: R\times (R/I) & \longrightarrow (R/I),\\ (r,s+I)& \longmapsto r \cdot(s+I):= rs+I. \end{alignat*}

So, $(R/I,+)$ under with this scalar multiplication is a left $R-$module.

However, $R/I$ is not always a ring, because $I$ is not a 2-sided ideal of $R$, so we can not define the quotient ring.

Are all the claims above correct? And could you please give some examples for this factor module?

Thanks.

Answer 1

could you please give some examples for this factor module?

In $R=M_2(F)$, for any field $F$, the subset of matrices whose right column is zeros forms a left ideal that is not a right ideal, and you can compute the quotient. If you like you can even use the field of two elements so that it is very small.

The quotient is a left $R$ module, but not a ring.