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 (ℤ4, +, ⋅), 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 (ℤn, +, ⋅) is a unit if and only if gcd(m,n) =1. |
Thus, every non-zero element in the ring (ℤp, +, ⋅) are units when p is a prime. For example, $\bar{1}, \bar{2}, \bar{3}, \bar{4}$ are the units in (ℤ5, +, ⋅).
| As every non-singular matrix is invertible, in the ring (Mn(ℝ), +, ⋅) every non-singular matrix is a unit. |
FAQs
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.
Answer: The zero is not a unit in a non-trivial ring. Because, non-zero elements in that ring do not exist.
