Cryptography Reference
In-Depth Information
If
F
(
x
)
∈ k
[
x
] is irreducible then the
k
[
x
]-ideal (
F
(
x
))
={
F
(
x
)
G
(
x
):
G
(
x
)
∈ k
[
x
]
}
is
a prime ideal.
An
R
-ideal
I
is
maximal
if every
R
-ideal
J
such that
I
⊆
J
⊆
R
is such that either
J
=
I
or
J
=
R
.
Lemma A.9.2
An R-ideal I is maximal if and only if R/I is a field (hence, a maximal
R-ideal is prime). If I is a maximal R-ideal and S
⊂
R is a subring then I
∩
S is a prime
S-ideal.
Proof
For the first statement see Theorem III.2.20 of [
271
]orSectionII.2of[
329
].
The second statement is proved as follows: let
I
be maximal and consider the injection
S
→
R
inducing
S
→
R/I
with kernel
J
=
S
∩
I
. Then
S/J
→
R/I
is an injective ring
homomorphism into a field, so
J
is a prime
S
-ideal.
of
R
-ideals is called an
ascend-
ing chain
. A commutative ring
R
is
Noetherian
if every ascending chain of
R
-ideals is
finite. Equivalently, a ring is Noetherian if every ideal is finitely generated. For more details
see Section VIII.1 of [
271
] or Section X.1 of [
329
].
Let
R
be a commutative ring. A sequence
I
1
⊂
I
2
⊂···
Theorem A.9.3
(Hilbert basis theorem) If R is a Noetherian ring thenR
[
x
]
is a Noetherian
ring.
Proof
See Theorem 1 page 13 of [
199
], Theorem VIII.4.9 of [
271
] Section IV.4 of [
329
],
or Theorem 7.5 of [
15
].
Corollary A.9.4
[
x
1
,...,x
n
]
is Noetherian.
A
multiplicative subset
of a ring
R
is a set
S
such that 1
k
∈
S
,
s
1
,s
2
∈
S
⇒
s
1
s
2
∈
S
.
The
localisation
of a ring
R
with respect to a multiplicative subset
S
is the set
S
−
1
R
={
r/s
:
r
∈
R,s
∈
S
}
with the equivalence relation
r
1
/s
1
≡
0. For more details see Chapter
3of[
15
], Section 1.3 of [
488
], Section I.1 of [
327
], Section II.4 of [
329
] or Section III.4 of
[
271
]. In the case
S
r
2
/s
2
if
r
1
s
2
−
r
2
s
1
=
R
∗
, we call
S
−
1
R
the
field of fractions
of
R
.Ifp is a prime ideal of
=
−
p is a multiplicative subset and the localisation
S
−
1
R
is denote
R
p
.
Lemma A.9.5
If R is Noetherian and S is a multiplicative subset of R then the localisation
S
−
1
R is Noetherian.
R
then
S
=
R
Proof
See Proposition 7.3 of [
15
] or Proposition 1.6 of Section X.1 of [
329
].
Aring
R
is
local
if it has a unique maximal ideal. If m is a maximal idea of a ring
R
then the localisation
R
m
is a local ring. It follows that
R
m
is Noetherian.
A.10 Vector spaces and linear algebra
The results of this section are mainly used when we discuss lattices in Chapter
16
. A good
basic reference is Curtis [
151
].
Let
n
as row vectors. We interchangeably use the words
k
be a field. We write vectors in
k
n
. The zero vector is
0
points
and
vectors
for elements of
k
=
(0
,...,
0). For 1
≤
i
≤
n
the
i
th unit vector is
e
i
=
(
e
i,
1
,...,e
i,n
) such that
e
i,i
=
1 and
e
i,j
=
0for1
≤
j
≤
n
and
j
=
i
.