24.
Rings
24.1.
Definitions and basic examples.
Definition.
A ring
R
is a set with two binary operations + (addition) and
·
(multiplication) satisfying the following axioms:
(A0)
R
is closed under addition
(A1) addition is associative
(A2) there exists 0
∈
R
s.t.
a
+ 0 = 0 +
a
for all
a
∈
R
(A3) for every
a
∈
R
there exists

a
∈
R
s.t.
a
+ (

a
) = (

a
) +
a
= 0
(A4) addition is commutative
(M0)
R
is closed under multiplication
(M1) multiplication is associative
(D1) distributivity on the left: (
a
+
b
)
c
=
ac
+
bc
for all
a, b, c
∈
R
(D2) distributivity on the right:
c
(
a
+
b
) =
ca
+
cb
for all
a, b, c
∈
R
Remark:
The axioms (A0)(A3) simply say that
R
is a group with respect
to addition. Axiom (A4) says that the group (
R,
+) is abelian.
Definition.
A ring
R
is called commutative
if multiplication is commuta
tive, that is,
ab
=
ba
for all
a, b
∈
R
.
Definition.
A ring
R
is a called a ring with 1
(or a ring with unity
) if there
exists 1
∈
R
s.t.
a
·
1 = 1
·
a
=
a
for all
a
∈
R
.
Definition.
A ring
R
is called a field
if
(i)
R
is commutative,
(ii)
R
is a ring with 1
(iii) for every
a
∈
R
, with
a
6
= 0, there exists
a

1
∈
R
such that
a
·
a

1
=
a

1
·
a
= 1
(iv) 1
6
= 0.
Examples:
1. Fields. The examples of fields we know so far are
Q
,
R
,
C
and
Z
p
where
p
is prime.
2. Commutative rings with 1, which are not fields. The examples we have
seen so far are
Z
and
Z
n
where
n
is nonprime. Another important class of
examples is given by polynomial rings:
Let
R
be a commutative ring with 1, and let
R
[
x
] denote the set of all
polynomials with coefficients in
R
. One can think of
R
[
x
] as the set of formal
1
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2
expressions
a
0
+
a
1
x
+
. . .
+
a
n
x
n
, with
a
i
∈
R
. Addition and multiplication
on
R
[
x
] are defined according to the “usual algebra rules”, e.g.
(
a
0
+
a
1
x
) + (
b
0
+
b
1
x
+
b
2
x
2
) = (
a
0
+
b
0
) + (
a
1
+
b
1
)
x
+
b
2
x
2
and
(
a
0
+
a
1
x
)
·
(
b
0
+
b
1
x
+
b
2
x
2
) =
a
0
b
0
+(
a
0
b
1
+
a
1
b
0
)
x
+(
a
0
b
2
+
a
1
b
1
)
x
2
+
a
1
b
2
x
3
.
It is a rather long but straightforward calculation to check that
R
[
x
] with
these operations becomes a ring with 1. The unity element of
R
[
x
] is the
constant polynomial 1 (where 1 is the unity element of
R
).
3. Commutative rings without 1. The basic examples are the rings
n
Z
where
n
≥
2 is a fixed integer.
4. Noncommutative rings. The basic examples are the matrix rings
Mat
n
(
F
)
where
F
is some field and
n
≥
2.
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 Winter '11
 PhanThuongCang
 Addition, Ring, Abelian group, Ring theory, additive inversion

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