Let \(R\) be a ring with unity and \(I\) be a two-sided ideal in \(R\text{.}\) The set of all cosets of \(R\) modulo \(I\) is denoted by \(R/I\text{.}\) Thus,
Since \(I\) is a subgroup of an abelian group \((R,+)\text{,}\)\(R/I\) is an abelian group. We define multiplication on \(R/I\) using multiplication on \(R\text{:}\)
We only show that if \(r+I=r^\prime+I\) then \((r+I)\cdot (s+I)=(r^\prime+I)(s+I)\text{.}\) As \(r+I=r^\prime+I\text{,}\) i.e., \(r-r^\prime\in I\text{,}\) and \(I\) is an ideal we have
\begin{equation*}
(r-r^\prime)s=rs-r^\prime s\in I.
\end{equation*}
In particular, \(rs+I=(r+I)\cdot (s+I)=(r^\prime+I)(s+I)=r^\prime s+I.\)
Verify that \(0+I\) is the additive identity of \(R/I\text{,}\) and \(1+I\) is the unity in \(R/I\text{.}\)
Verify that the addition and multiplication on \(R/I\) satisfy distributive laws given in Definition 1.1.1(4).
Hence \(R/I\) is a ring with unity.
Lemma3.1.2.
The canonical map \(\pi\colon R\to R/I\) defined by
\begin{equation*}
r\mapsto r+I
\end{equation*}
is a ring epimorphism with kernel, \(\ker(\pi)=I\text{.}\)
Convention3.1.3.
Let \(R\) be a ring and let \(I\) be a two-sided ideal of \(R\text{.}\) We write \(r\equiv s\mod I\) for \(r-s\in I\text{.}\)
