Quotient Rings

Quotient Rings

Recall from the Ideals of Rings page that if $(R, +, *)$ is a ring then $I$ is said to be an ideal of $(R, +, *)$ if:

• 1) $(I, +)$ is a subgroup of $(R, +)$.
• 2) For all $r \in R$ and for all $i \in I$ we have that $r * i \in I$ and $I * r \in I$. In other words, for all $r \in R$ we have that $rI \subseteq I$ and $Ir \subseteq I$.

Like discussed previously, with ideal subrings, we will be able to general the concept of a quotient group (which required us to have normal subgroup) to quotient rings.

Let $(R, +, *)$ be a ring and let $(I, +, *)$ be an ideal. Define an equivalence relation $\sim$ for all $a, b \in R$ by:

(1)
\begin{align} \quad a \sim b \quad \Leftrightarrow a - b \in I \end{align}

We denote the equivalence classes of these rings be denoted by:

(2)
\begin{align} \quad [a] = a + I = \{ a + i : i \in I \} \end{align}

We let $R / I$ be the set of all of these equivalence classes and define addition and multiplication for all $(a + I), (b + I) \in R / I$ by:

(3)
\begin{align} \quad (a + I) + (b + I) = (a + b) + I \end{align}
(4)
\begin{align} \quad (a + I) * (b + I) = (a * b) + I \end{align}

It is easy to show that then $R/I$ is a ring with these operations, and this ring is given a special name.

 Theorem 1: Let $(R, +, *)$ be a ring and let $(I, +, *)$ be an ideal. Then the set $R / I$ with the operations defined above is called the Quotient Ring of $R$ by $I$.