Units in a Ring: Definition, Examples, How to Find

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.

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

1. In (ℤ, +, ⋅), 1 and -1 are the only units.
2. Each non-zero element in the rings (ℚ, +, ⋅) and (ℝ, +, ⋅) are units.
3. 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 (ℤ_p, +, ⋅) 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.

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