Abstract Algebra begins with the group. Groups are a general and powerful structure which
encompass a wide range of objects: the integers, polynomials, matrices, modular arithmetic,
functions, geometric transformations, and much more. After you’ve spent some time
learning about groups, you’re ready to move on to more specific structures -- groups with
extra features. Today, we’ll talk about additive groups which have a second operation:
multiplication. We call these rings...
Before we give the textbook definition of a ring, let’s see some examples, which
will motivate the idea. Consider these four sets: the integers… the real numbers…
2-by-3 matrices with real entries… and polynomials with complex coefficients. In arithmetic you
learn about four operations: addition, subtraction, multiplication and division. Let’s see which
operations are available for each of these sets.
For the integers “Z”, you can freely add, subtract and multiply any two integers and
you’ll get another whole number. But notice that you can’t always divide.
For example, 1 divided by 2 is ½ , which is NOT an integer - it’s a fraction and
is not in the set of integers. So for the integers, we only have 3 of the
4 operations from arithmetic.
For the real numbers, you can add, subtract,
multiply and divide any two real numbers, with the standard caveat that you cannot divide
by zero.
So the real numbers come equipped with all 4 operations.
For the 2-by-3 matrices, you can add and subtract
any two such matrices, but you cannot multiply or divide.
This is because you can only multiply two matrices when the number of columns in the first matrix
is the same as the number of rows in the second matrix.
For example, you can multiply a 3-by-5 matrix by a 5-by-2 matrix because
the width of the first is equal to the height of the second.
And to divide one matrix “A” by another “B”, we have to multiply “A” by the
inverse of “B”, and not every matrix is invertible.
So this set only has 2 of the 4 operations.
And finally, you can add, subtract and multiply
any two polynomials, and you’ll get another polynomial.
But if you divide one by another, you usually get a “rational function”, which is NOT
a polynomial. The elements in the set of polynomials only
have 3 of the 4 operations.
These examples show that, depending on the set, you’ll have different operations from
arithmetic available to you. And while in arithmetic you learn that there are four operations,
in abstract algebra we think about things a little differently. Here, there are really
only two operations - addition and multiplication. Let’s see why...
2 - 7 is the same as 2 + (-7) So here, subtraction is the same as adding
a negative integer.
Also, 3 - (-5) is the same as 3 + 5, so in
this example subtracting -5 is the same as adding the opposite, which is 5.
These examples show that subtraction is the same as adding an “opposite” element.
But we use the word “inverse” instead of “opposite.”
So in abstract algebra, “subtraction” is the same as “addition” with “inverses”.
What’s the benefit for this change in thinking? Why not continue to talk about “addition”
and “subtraction”?
One reason is that in abstract algebra, the elements may not
be numbers. For example, the symmetric group on three elements is a group with 6 elements,
where each element is a permutation. What does it mean to subtract the permutation 2-3-1?
What is the negative of 2-3-1? It makes more sense to talk about combining one element
with the inverse of another.
Similarly, “division” is “multiplication” with “inverses”.
For example, 3 divided by 5 is the same as 3 times ⅕.
And ⅔ divided by 4/7 can be written as ⅔ times 7/4.
We see for multiplication, “inverse” means “reciprocal” while for addition, “inverse”
means the opposite. To keep things clear, we’ll often use the
phrases “additive inverses” and “multiplicative inverses.”
We already have a word for a set where you can add any two elements, every element has
an additive inverse, and you have the required rules from arithmetic: we call this a group.
So in our four sets, all four are groups under addition. Better still, they are all commutative groups.
But three of them also have multiplication,
and one of them has multiplication with multiplicative inverses.
We need names to describe sets with these additional features.
When a commutative group under addition also has multiplication, we call it a ring.
Let’s see the official definition.
A ring is a set of elements “R” with two operations: addition and multiplication.
For both operations the set is “closed.” This means if you add any two elements in
“R”, you get another element in “R”.
Similarly, if you multiply any two elements in “R”,
you get a third element in “R”. For addition, the elements form an abelian group.
For multiplication, there are fewer requirements.
Like addition, multiplication has the associative property.
But that’s it.
We don’t require multiplication to be commutative.
We also don’t require elements to have multiplicative inverses.
There’s still some debate on whether or
not a ring should have an identity element 1 for multiplication.
Because of this, you’ll often hear people talk about “a ring with identity.” This
just means the ring has an element “1”.
And finally, these two operations are linked
by the distributive properties.
This is the official definition of a ring.
There are additional names for rings depending on
how close it comes to being a group under multiplication.
For instance, while all rings are abelian under addition, they are not required
to be commutative under multiplication. The two-by-two matrices with real entries are
a great example of a ring that’s not commutative.
But if a ring IS commutative, we call it a
“commutative ring.”
Also, rings are not required to have multiplicative inverses. If a ring DOES have inverses under
multiplication, then we call it a “division ring.” This is a very descriptive name,
because the presence of multiplicative inverses allows us to define division. Division rings
may or may not be commutative.
A classic example of a non-commutative division ring is the
quaternions -- a quirky 4-dimensional analog to the complex numbers.
But if a division ring IS commutative, then
we call it a field.