In a ring R, an element is called a unit if its multiplicative inverse exists. That is, if ab=I (the unity in R), then both a and b are units. In this post, we will study the units of a ring with its definition, examples, and how to find it.

Table of Contents

## Definition of Units in a Ring

Let R be a non-trivial ring with unity I. An element a ∈ R is called a unit if there exists an element b in R such that

a⋅b = b⋅a = I.

The element b is called the multiplicative inverse of a.

## Examples of Units

- In (ℤ, +, ⋅), 1 and -1 are the only units.
- Each non-zero element in the rings (ℚ, +, ⋅) and (ℝ, +, ⋅) are units.
- In (ℤ
_{4}, +, ⋅), the units are $\bar{1}$ and $\bar{3}$. This is because $\bar{1} \cdot \bar{1} = \bar{1}$ and $\bar{3} \cdot \bar{3}=\bar{1}$.

**Must Read:** An Introduction to Ring Theory

## Theorems

**Theorem 1:** The multiplicative inverse of an element of a ring with unity is unique.

**Proof:**

Let R be a ring with unity and a ∈ R.

Suppose that there are two multiplicative inverses b and c of a. Then by definition, we must have that

a⋅b = b⋅a = I

a⋅c = c⋅a = I

We need to show that b=c.

Now, b⋅(a⋅c) = (b⋅a)⋅c as the multiplication is associative. ⇒ b⋅I = I⋅c ⇒ b = c. |

This proves that the multiplicative inverse of a is unique. As a is arbitrary, we complete the proof of the theorem.

**Theorem 2:** The zero element of a ring R is not a unit.

**Proof:**

If R is a ring without unity, then no element is a unit, so is the zero element. Thus assume that R is a ring with unity I.

If 0 is a unit, then its multiplicative inverse 0^{-1} exists, and by definition we have

0⋅0^{-1} = 0^{-1}⋅0 = I

On the other hand, by the property of the zero element, 0⋅0^{-1} = 0^{-1}⋅0 = 0. This proves that I=0.

⇒ a⋅I = a⋅0 (here we consider a to be a non-zero element)

⇒ a = 0.

So we arrive at a contradiction. In other words, 0^{-1} does not exist. This proves that 0 is not a unit in R.

## How to Find Units in a Ring

In this section, we will learn how to find units in various rings with examples.

An element $\bar{m}$ of the ring (ℤ_{n}, +, ⋅) is a unit if and only if gcd(m,n) =1. |

Thus, every non-zero element in the ring (ℤ_{p}, +, ⋅) are units when p is a prime. For example, $\bar{1}, \bar{2}, \bar{3}, \bar{4}$ are the units in (ℤ_{5}, +, ⋅).

As every non-singular matrix is invertible, in the ring (M_{n}(ℝ), +, ⋅) every non-singular matrix is a unit. |

## FAQs

**Q1: What are units in a ring?**

Answer: An element with a multiplicative inverse is called a unit in a ring R. That is, a ∈ R is called a unit if ∃ b in R such that a⋅b = b⋅a = I. For example, -1 is a unit in the ring (ℤ, +, ⋅) as (-1)⋅(-1) = 1, the unity in ℤ.

**Q2: Is zero a unit in a ring?**

Answer: The zero is not a unit in a non-trivial ring. Because, non-zero elements in that ring do not exist.

This article is written by Dr. T. Mandal, Ph.D in Mathematics. On Mathstoon.com you will find Maths from very basic level to advanced level. Thanks for visiting.