Accumulation of Definitions, Theorems, Corollaries, and Propositions provided by Professor Robert Boltje
in 111B: Ring Theory, Spring 2010. If any errors are found, contact Gary Kirby at [email protected]
Chapter 1: Rings and Fields
[1.1] Definition
: A
Ring
(
R,
+
,
×
) is a set
R
with two binary opertaions called addition and multiplication
s.t.
1. (
R,
+) is an abelian group
2. (
R,
×
) is associative
3. (Distributive Laws)
(a)
a
(
b
+
c
) = (
ab
) + (
ac
)
(b) (
a
+
b
)
c
= (
ac
) + (
bc
)
Types/Cases of Rings
:
1. If (
R,
+
,
×
) is multiplicatively commutative, than R is a
Commutative ring
2. If (
R,
+
,
×
) has a multiplicative identity, 1
R
, than R is a
Unitary Ring
3. If (
R,
+
,
×
) has a multiplicative inverse for all nonzero elements, than R is a
Division Ring
, i.e.
∪
(
R
) =
R
\{
0
}
4. If (
R,
+
,
×
)is a Division Ring, that is commutative, than R is a
Field
5. If (
R,
+
,
×
) has a multiplicative identity with 0
R
= 1
R
, no 0 divisors, and commutative, than R is a
Integral Domain
6. If (
R,
+
,
×
) is a Integral Domain, and every nonzero element is a unit, than R is a
Field
[1.8] Definition
: Let R and S be rings. A function
ϕ
:
R
→
S
is called a
ring homomorphism
if
ϕ
(
a
+
b
) =
ϕ
(
a
) +
ϕ
(
b
)
and
ϕ
(
ab
) =
ϕ
(
a
)
ϕ
(
b
)
∀
a, b
∈
R.
If R and S are unitary rings and if additionally
ϕ
(1
R
) = 1
S
, than
ϕ
is a
unitary ring homomorphism
. A
homomorphism
ϕ
: R
→
S is called an isomorphism if
ϕ
is bijective, thus R
∼
= S.
[1.11] Definition
: Let R be a ring.
A subset S
≤
R is called a subring of R if S is a subgroup under
addition and if S is closed under multiplication. In this case, S is again a ring with induced addition and
multplication. If R is unitary and 1
R
∈
S, then 1
R
is a unity in S and called a unitary subring. Let F be a
field. A subring E of F is again a field with the induced addition and multiplication is than called a subfrield
of F. (Automatically 1
E
= 1
R
)
Chapter 2: Integral Domains
[2.2] Definition
: A nonzero element of a ring R, is called a zerodivisor if there exists an element
b
∈
R
with
ab
= 0 or
ba
= 0.
[2.3] Proposition
: Let 1
< n
∈
N
and let
a
∈
{
1
, ..., n

1
}
1
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1.
a
is a zero divisor in
Z
n
⇔
gcd
(
a, n
)
>
1
2.
a
is a unit in
Z
n
⇔
gcd
(
a, n
) = 1
[2.4] Definition
:
We say that the cancellation laws hold in a ring R if
∀
a, b
∈
R with
a
= 0 has:
ab
=
ac
⇒
b
=
c
and
ba
=
ca
⇒
b
=
c.
[2.5] Proposition
: The cancellation laws hold in a ring R
⇔
R has no zerodivisors.
[2.8] Proposition
: Let R be a finite ring.
R
is a field
⇔
R
is an integral domain.
[2.9] Definition
: Let R be a ring, If there exists n
∈
N
with
na
= 0
∀
a
∈
R then the smallest such
n
is
called the characterisic of R,
char
(
R
). Otherwise, one sets
char
(
R
) = 0.
[2.10] Proposition
:
If R is a unitary ring then
char
(
R
) =
min
{
n
∈
N

(
n
)(1) = 0
}
if it exists and
char
(
R
) = 0 otherwise.
[2.11] Proposition
: The characteristic of an integral domain R is either a prime or 0.
Chapter 3: Fermat’s and Euler’s Thereom
[3.3] Theorem (Euler)
: Let 1
< n
∈
N
and let a
∈
Z
with
gcd
(
a, n
) = 1. Then
a
ϕ
(
a
)
≡
1
modn
.
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 Spring '10
 jim
 Ring, Integral domain, ring homomorphism, Ring theory

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