Cryptography Reference
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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 .
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