2Rings
IB Groups, Rings and Modules
2.1 Definitions and examples
We now move on to something completely different — rings. In a ring, we are
allowed to add, subtract, multiply but not divide. Our canonical example of a
ring would be Z, the integers, as studied in IA Numbers and Sets.
In this course, we are only going to consider rings in which multiplication
is commutative, since these rings behave like “number systems”, where we can
study number theory. However, some of these rings do not behave like
Z
. Thus
one major goal of this part is to understand the different properties of
Z
, whether
they are present in arbitrary rings, and how different properties relate to one
another.
Definition (Ring). A ring is a quintuple (
R,
+
, ·,
0
R
,
1
R
) where 0
R
,
1
R
∈ R
,
and +, · : R × R → R are binary operations such that
(i) (R, +, 0
R
) is an abelian group.
(ii) The operation · : R × R → R satisfies associativity, i.e.
a ·(b · c) = (a · b) · c,
and identity:
1
R
· r = r · 1
R
= r.
(iii) Multiplication distributes over addition, i.e.
r
1
· (r
2
+ r
3
) = (r
1
· r
2
) + (r
1
· r
3
)
(r
1
+ r
2
) · r
3
= (r
1
· r
3
) + (r
2
· r
3
).
Notation. If
R
is a ring and
r ∈ R
, we write
−r
for the inverse to
r
in (
R,
+
,
0
R
).
This satisfies r + (−r) = 0
R
. We write r − s to mean r + (−s) etc.
Some people don’t insist on the existence of the multiplicative identity, but
we will for the purposes of this course.
Since we can add and multiply two elements, by induction, we can add and
multiply any finite number of elements. However, the notions of infinite sum and
product are undefined. It doesn’t make sense to ask if an infinite sum converges.
Definition (Commutative ring). We say a ring
R
is commutative if
a · b
=
b · a
for all a, b ∈ R.
From now onwards, all rings in this course are going to be commutative.
Just as we have groups and subgroups, we also have subrings.
Definition (Subring). Let (
R,
+
, ·,
0
R
,
1
R
) be a ring, and
S ⊆ R
be a subset.
We say
S
is a subring of
R
if 0
R
,
1
R
∈ S
, and the operations +
, ·
make
S
into a
ring in its own right. In this case we write S ≤ R.
Example. The familiar number systems are all rings: we have
Z ≤ Q ≤ R ≤ C
,
under the usual 0, 1, +, ·.
Example. The set
Z
[
i
] =
{a
+
ib
:
a, b ∈ Z} ≤ C
is the Gaussian integers, which
is a ring.
We also have the ring Q[
√
2] = {a + b
√
2 ∈ R : a, b ∈ Q} ≤ R.
We will use the square brackets notation quite frequently. It should be clear
what it should mean, and we will define it properly later.
In general, elements in a ring do not have inverses. This is not a bad thing.
This is what makes rings interesting. For example, the division algorithm would
be rather contentless if everything in
Z
had an inverse. Fortunately,
Z
only has
two invertible elements — 1 and −1. We call these units.
Definition (Unit). An element
u ∈ R
is a unit if there is another element
v ∈ R
such that u · v = 1
R
.
It is important that this depends on
R
, not just on
u
. For example, 2
∈ Z
is
not a unit, but 2 ∈ Q is a unit (since
1
2
is an inverse).
A special case is when (almost) everything is a unit.
Definition (Field). A field is a non-zero ring where every
u 6
= 0
R
∈ R
is a unit.
We will later show that 0
R
cannot be a unit except in a very degenerate case.
Example. Z is not a field, but Q, R, C are all fields.
Similarly, Z[i] is not a field, while Q[
√
2] is.
Example. Let
R
be a ring. Then 0
R
+ 0
R
= 0
R
, since this is true in the group
(R, +, 0
R
). Then for any r ∈ R, we get
r · (0
R
+ 0
R
) = r · 0
R
.
We now use the fact that multiplication distributes over addition. So
r · 0
R
+ r · 0
R
= r · 0
R
.
Adding (−r · 0
R
) to both sides give
r · 0
R
= 0
R
.
This is true for any element
r ∈ R
. From this, it follows that if
R 6
=
{
0
}
, then
1
R
6= 0
R
— if they were equal, then take r 6= 0
R
. So
r = r ·1
R
= r · 0
R
= 0
R
,
which is a contradiction.
Note, however, that
{
0
}
forms a ring (with the only possible operations
and identities), the zero ring, albeit a boring one. However, this is often a
counterexample to many things.
Definition (Product of rings). Let
R, S
be rings. Then the product
R × S
is a
ring via
(r, s) + (r
0
, s
0
) = (r + r
0
, s + s
0
), (r, s) · (r
0
, s
0
) = (r · r
0
, s · s
0
).
The zero is (0
R
, 0
S
) and the one is (1
R
, 1
S
).
We can (but won’t) check that these indeed are rings.
Definition (Polynomial). Let
R
be a ring. Then a polynomial with coefficients
in R is an expression
f = a
0
+ a
1
X + a
2
X
2
+ ··· + a
n
X
n
,
with a
i
∈ R. The symbols X
i
are formal symbols.
We identify f and f + 0
R
· X
n+1
as the same things.
Definition (Degree of polynomial). The degree of a polynomial
f
is the largest
m such that a
m
6= 0.
Definition (Monic polynomial). Let
f
have degree
m
. If
a
m
= 1, then
f
is
called monic.
Definition (Polynomial ring). We write
R
[
X
] for the set of all polynomials
with coefficients in
R
. The operations are performed in the obvious way, i.e. if
f
=
a
0
+
a
1
X
+
···
+
A
n
X
n
and
g
=
b
0
+
b
1
X
+
···
+
b
k
X
k
are polynomials,
then
f + g =
max{n,k}
X
r=0
(a
i
+ b
i
)X
i
,
and
f · g =
n+k
X
i=0
i
X
j=0
a
j
b
i−j
X
i
,
We identify
R
with the constant polynomials, i.e. polynomials
P
a
i
X
i
with
a
i
= 0 for
i >
0. In particular, 0
R
∈ R
and 1
R
∈ R
are the zero and one of
R
[
X
].
This is in fact a ring.
Note that a polynomial is just a sequence of numbers, interpreted as the
coefficients of some formal symbols. While it does indeed induce a function in
the obvious way, we shall not identify the polynomial with the function given by
it, since different polynomials can give rise to the same function.
For example, in
Z/
2
Z
[
X
],
f
=
X
2
+
X
is not the zero polynomial, since its
coefficients are not zero. However,
f
(0) = 0 and
f
(1) = 0. As a function, this is
identically zero. So f 6= 0 as a polynomial but f = 0 as a function.
Definition (Power series). We write
R
[[
X
]] for the ring of power series on
R
,
i.e.
f = a
0
+ a
1
X + a
2
X
2
+ ··· ,
where each
a
i
∈ R
. This has addition and multiplication the same as for
polynomials, but without upper limits.
A power series is very much not a function. We don’t talk about whether
the sum converges or not, because it is not a sum.
Example. Is 1
−X ∈ R
[
X
] a unit? For every
g
=
a
0
+
···
+
a
n
X
n
(with
a
n
6
= 0),
we get
(1 − X)g = stuff + ···− a
n
X
n+1
,
which is not 1. So
g
cannot be the inverse of (1
− X
). So (1
− X
) is not a unit.
However, 1 − X ∈ R[[X]] is a unit, since
(1 − X)(1 + X + X
2
+ X
3
+ ···) = 1.
Definition (Laurent polynomials). The Laurent polynomials on
R
is the set
R[X, X
−1
], i.e. each element is of the form
f =
X
i∈Z
a
i
X
i
where
a
i
∈ R
and only finitely many
a
i
are non-zero. The operations are the
obvious ones.
We can also think of Laurent series, but we have to be careful. We allow
infinitely many positive coefficients, but only finitely many negative ones. Or
else, in the formula for multiplication, we will have an infinite sum, which is
undefined.
Example. Let
X
be a set, and
R
be a ring. Then the set of all functions on
X
,
i.e. functions f : X → R, is a ring with ring operations given by
(f + g)(x) = f (x) + g(x), (f · g)(x) = f(x) · g(x).
Here zero is the constant function 0 and one is the constant function 1.
Usually, we don’t want to consider all functions
X → R
. Instead, we look at
some subrings of this. For example, we can consider the ring of all continuous
functions
R → R
. This contains, for example, the polynomial functions, which
is just R[X] (since in R, polynomials are functions).