Units (Multiplicatively Invertible Elements) in Rings

# Units (Multiplicatively Invertible Elements) in Rings

 Definition: Let $(R, +, *)$ be a ring with identity $1$. An element $a \in R$ is said to be a Unit or Multiplicatively Invertible Element if there exists an element $b \in R$ such that $a * b = 1$ and $b * a = 1$. If such a $b$ satisfies this condition, then we sometimes denote it by $b = a^{-1}$.

For example, consider the ring $(\mathbb{R}, +, *)$ of real numbers under standard addition $+$ and standard multiplication $*$ where $1$ is the identity of $*$. For every $a \in \mathbb{R} \setminus \{ 0 \}$ we can define $a^{-1} = \frac{1}{a} \in \mathbb{R}$ so that then:

(1)
\begin{align} \quad a * a^{-1} = a * \frac{1}{a} = \frac{a}{a} = 1 \quad \mathrm{and} \quad a^{-1} * a = \frac{1}{a} * a = \frac{a}{a} = 1 \end{align}

As we can see, the ring $(\mathbb{R}, +, *)$ has multiplicative inverses for every nonzero element in $\mathbb{R}$ and so every nonzero element in $\mathbb{R}$ is multiplicatively invertible.

Now consider the ring $(\mathbb{Z}, +, *)$ is integers with respect to standard addition $+$ and standard multiplication $*$ where $1$ is the identity of $*$. Let $a, b \in \mathbb{Z}$ and consider the equation:

(2)
\begin{align} \quad a * b = 1 \end{align}

We know from our experience with the set of integers that we must have that either $a = b = 1$ or $a = b = -1$ to satisfy the equation above. Therefore $1, -1 \in \mathbb{Z}$ are multiplicatively invertible elements whose multiplicative inverses are respectively themselves. Note that if $a \neq \pm 1$ then there exists no $b \in \mathbb{Z}$ such that $a * b = 1$. For example, $a = 2 \neq \pm 1$ would have a multiplicative inverse of $b = \frac{1}{2}$ but then $b \not \in \mathbb{Z}$.

The ring of integers $(\mathbb{Z}, +, *)$ above is a prime example of a ring that contains a finite number of multiplicatively invertible elements, while the ring $(\mathbb{R}, +, *)$ of real numbers from earlier is a prime example of a ring that contains an infinite number of multiplicatively invertible elements.