Division Rings

We will now look at another type of ring called division rings. These are rings with identity that also have inverses for the operation $*$ except for the identity $0$ of $+$.

Definition: Let $(R, +, *)$ be a ring and let $1$ be the multiplicative identity. Then $(R, +, *)$ is a Division Ring if for all $x \in R$, $x \neq 0$ (where $0$ is the identity for $+$) we have that there exists an $x^{-1} \in R$ such that $x * x^{-1} = 1$ and $x^{-1} * x = 1$.

Once again, many of the basic rings that we have already seen are also division rings. For example, $(\mathbb{R}, +, *)$ is a division ring since for every real number $x \neq 0$ we have that $\displaystyle{\frac{1}{x} \in \mathbb{R}}$ is such that:

(1)

\begin{align} \quad x * \frac{1}{x} = 1 \quad \mathrm{and} \quad \frac{1}{x} * x = 1 \end{align}

There are of course rings that are not division rings. For example, consider the ring $(\mathbb{Z}, +, *)$. This is indeed a ring with identity but it is not a division ring. Take the element $2 \in \mathbb{Z}$. There exists no element $z \in \mathbb{Z}$ such that $2z = 1$. I.e., there is no integer solution to the equation $2z = 1$. So $(\mathbb{Z}, +, *)$ is not a division ring.