Units in a Ring

# Units in a Ring

 Definition: Let $(R, +, \cdot)$ be a ring with multiplicative identity $1$. A Unit or (Multiplicatively) Invertible Element of $R$ is an element $a \in R$ for which there exists a $b \in R$ with $a \cdot b = 1$ and $b \cdot a$ = 1. The Set of All Units in $R$ is denoted by $R^{\times}$.

Let's look at some examples of units in a ring.

## Example 1

Consider the standard ring of real numbers $(\mathbb{R}, +, \cdot)$. Let $a \neq 0$ and let $\displaystyle{b = \frac{1}{a} \in \mathbb{R}}$. Then:

(1)
\begin{align} \quad a \cdot b = a \cdot \frac{1}{a} = 1 \quad \mathrm{and} \quad b \cdot a = \frac{1}{a} \cdot a = 1 \end{align}

On the otherhand, if $a = 0$ and $b \in R$ then $0 \cdot b = 1$ has no solutions. So the set of units in $\mathbb{R}$ is the set of nonzero real numbers.

## Example 2

Consider the ring $(M_{nn}, +, \cdot)$ of $n \times n$ matrices with entries in $\mathbb{R}$. The multiplicative identity of $M_{nn}$ is the $n \times n$ identity matrix, $I_n$. If $A \in M_{nn}$ we note that $AB = I_{n} = BA$ if and only if $A$ is invertible.

Therefore the set of units in $M_{nn}$ is the set of $n \times n$ invertible matrices, i.e., the set of matrices with nonzero determinant.