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.

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## 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.