Observation

Lets look at the following examples with rational numbers

• $5 * (\frac{1}{5}) = 1$

We say element 5 $\in \mathbb{Q}$ has a multiplicative inverse. Note that $(\frac{1}{5})\in \mathbb{Q}$. Additionally, note that the multiplicative inverse of five is one fifth, and the multiplicative inverse of one fifth is five.

• $4 * (\frac{1}{4}) = 1$

We say element $(\frac{1}{4}) \in \mathbb{Q}$ has a multiplicative inverse. Note that $4 \in \mathbb{Q}$

• $0 * (?) = 1$

Since $(\frac{1}{0})$ is not defined, $0$ does not have a multiplicative inverse. ie there is no number that you can multiply by 0 to get 1.

Definition : Units

Let $x \in R$ where R is a ring. We say that $x$ is a unit, if there exists $y \in R$ such that $x * y = 1$ and $y * x = 1$

Units In $\mathbb{Z}$

What are the integers in $\mathbb{Z}$ that are units?

From the definition of units, we need two elements $x,y \in \mathbb{Z}$ such that $x \times y =1$

This implies that $x = \frac{1}{y}$.

$\frac{1}{y} \in \mathbb{Z}$ only when $y=\pm1$

The units in $\mathbb{Z}$ are therefore $(1,-1)$ . These are the integer solutions to $x^2 = 1$.