Factor rings

Let $R$ be a ring and let $I$ be an ideal of $R$.

• Definition of a coset

A coset of $I$ in $R$ is a set $r + I = \{r+a \ | \ a \in I\}$ where $r \in R$.

• Theorem (equality of cosets)

The cosets of $I$ in $R$ partition the ring $R$. In particular

$r_1 + I = r_2 + I \mbox{ if and only if } r_1 - r_2 \in I.$

• Definition of the factor ring

The set of cosets of $I$ in $R$ is a ring under

$\mbox{addition: } r_1 + I + r_2 + I = r_1 + r_2 + I$;
$\mbox{multiplication } (r_1 + I)(r_2 + I) = r_1r_2 + I.$

This ring is denoted $R/I$ and is called the factor ring of $R$ by $I$.

• First Isomorphism Theorem

Let $R, S$ be rings and let $\varphi : R \to S$ be a ring homomorphism. Then $R/\mbox{Ker}\,\varphi \cong \mbox{Im}\,\varphi .$

Next section on Maximal ideals