In a ring R, every element has an additive inverse, but not every element
will have a multiplicative inverse. Put another way, for a ring R, the
elements form a group under addition, but the nonzero elements may not form a
group under multiplication. However, there may be elements in the
ring which do have multiplicative inverses. We call these elements Units,
and they form a group under multiplication:
the group of units in a ring.
Let R be a ring. An element X is called a unit if there's an element Y
where x times y equals 1 and y times x equals 1. This is the notation for the
inverse of X. The set of all units for the ring R is written with a little
multiplication symbol up above. Let's first see a few examples before we talk
about the units abstractly.
The integers Z are a commutative ring. The only numbers
with the multiplicative inverse are 1 and negative 1. For every other nonzero
integer, the inverse would be a fraction, not an integer.
For example, the inverse of 3 is 1/3, which is not an integer. So the units for
this ring is a group with only 2 elements: 1 and negative 1. Here is a
group multiplication table for these units.
Next, let's look at the integers mod 12.
To find the units, let's look at the multiplication table for the non-zero
elements 1 through 11. We can find the units by looking for rows or columns
that contain the multiplicative identity 1. The only rows and columns that contain
a 1 are 1, 5, 7, and 11.
These are the 4 units for the ring of integers mod 12.
You can check that this is a group by looking at the group multiplication
table for these 4 units. This table only contains the numbers 1, 5, 7, and 11, so
it's closed under multiplication. There's a 1 in every row in column so each
element has an inverse. So the units for the integers mod 12 form a group of
order 4. Now that we've seen a couple of examples,
let's talk about the units more generally. In the previous examples, we
saw that the units form a group under multiplication.
Let's now see why this is always true.
Let R be any ring. Let's look at the set
of units and show that it's a group. To do this, we need to check that all the
group axioms hold. First, the set of units does contain the identity element 1,
since 1 is its own inverse under multiplication. So this set has an
identity. We inherit associativity from the ring R, so we can check that off. Next,
by definition, every element X in this set has an inverse Y, so we can mark off
the inverse axiom. This leaves closure. In other words, if x
and y are elements in this set, is their product also in the set? That is, if X has
an inverse and Y has an inverse, does X times y also have an inverse? Indeed, it
does. The inverse of x times y is y inverse times X inverse. Since x times y
has an inverse, it must also be in the set of units. So we have closure. This
shows the units do indeed form a group under multiplication.
Let's return to the example of the integers Z.
This ring has two units: 1 and negative 1. If you
multiply any integer by a unit, you get what's called an associate. For example,
if you multiply the integer 2 by the units, you get 2 and negative 2. These
integers are associates. Similarly, 3 and negative 3 are associates, as are 4 and
negative 4,and so on. The ideas of units and associates are needed for a more
precise discussion of factoring numbers.
Most people know you can factor any
integer greater than 1 uniquely into a product of primes. This is known as a
fundamental theorem of arithmetic. We don't include one, because it's a unit,
and units aren't prime and do not have a unique factorization. For example, we can
factor the integer 1 as 1 times 1, 1 times 1 times 1, and so on...but what about
the negative integers? Do they have a unique factorization? They do, but you
need to use units and associates to describe it. For example, negative 12 can
be written as 2 times 2 times negative 3. It can also be written as negative 2
times negative 3 times negative 2, or as 2 times negative 2 times 3.
We can even factor it as negative 1 times 1 times 2 times 2 times 3.
But these factorizations are all related,
because 2 and negative 2 are associates, and 3 and negative 3 are also associates.
The last factorization just includes a few units thrown in. You can absorb these
units into the first factor. If we reorder the factorizations
so the associates line up with one another, we see the factorizations are
all the same - the only difference being the terms are associates of one another.
We often say the factorizations are unique up to order and associates. This
is the fundamental theorem of arithmetic generalized to all integers. It applies
to every integer except 0 and the 2 units 1 and negative 1. The problem of
finding the units in a ring is a major topic in number theory. This is because
the integers and the rational numbers have been generalized into extensions
call the ring of integers and number fields; and many problems in number
theory require an understanding of how many units there are and what is a
structure of this group. Describing the group of units is a difficult problem
that is still being researched today. As a final example, look at the ring of 2 by
2 matrices with integer entries. Most of these matrices do not have an inverse
under multiplication. For example, the inverse of the matrix 1
2 3 4 is the matrix negative 2 1 3 halves and negative 1/2. This matrix is
not in the ring, because two of the entries are fractions. But look at the
matrix 2 7 1 4. The inverse of this matrix is 4 negative 7 negative 1 2
which is in the ring. Their product is the identity matrix, regardless of the
order, so this matrix is a unit. In fact this ring has an infinite group of units:
the set of all matrices with determinant 1 or negative 1. We won't prove this here,
but it's a very rewarding challenge to try to convince yourself this is true.
Here's a fun problem for you to discuss: look at the integers mod n from N equals
2 through 50. For each ring, find all the units and then, compute the
percentage of elements that are units. Which ring has a smallest percentage of units?
Next, compare this to the percentage of subscribers who support us on Patreon.
One of these fractions is way too small. You should do something about that.....