Chapter 1 Definitions Flashcards
Divides
a|b ⇔ there exists a c ∈ Z s.t b = a c .
(a) multiples
(a) = aZ = { ka | k∈ Z}
An Ideal
An ideal in a commutative ring R is a subset I of R satisfying:
(i) 0 ∈ I
(ii) a, b∈ I ⇒ a±b ∈ I
(iii) a ∈ I , r ∈ R ⇒ ra ∈ I
Notation: I *triangle* R
Principle ideal
The set of all multiples of a fixed element a of R is the principle ideal (a) or aR.
a generates (a).
Associates of a
Associates of a are the elements,
b = u a
where u is a unit of R.
[Also generators of (a)]
A Principle Ideal Domain
A PID is a non-zero, commutative ring such that:
(i) ab=0 ⇒ a=0 or b=0,
(ii) Every ideal in R is principle.
The greatest common divisor (gcd)
The gcd of a,b∈Z is the integer d≥0 satisfying:
(i) d|a and d|b
(ii) c|a and c|b ⇒ c|d
(iii) d=au+bv where u,v ∈ Z (Besout’s identity)
a,b Coprime
a,b are coprime if gcd(a,b)=1
Ideal of a finite sequence
Ideal of a finite sequence
I=(a1, a2, … ,an) = { k1a1, k2a2, … ,knan | Ki ϵ Z }
is:
I=(d) where d = gcd(a1, a2, … ,an).
(a1, a2, …,an) are pairwise coprime
(a1, a2, …,an) are pairwise coprime if gcd(ai, aj) = 1 for i≠j.
Prime and composite numbers
A prime number is a an integer P>1 whose only divisors are ±1 and ±P.
Integers which are not prime are called composite.
ordp(n)
ordp(n) = e ⇔ pe | n and pe+1 -|- n
An Integral Domain ID
A non-zero, commutative, ring, R, is an Integral Domain if for a,b in Z, ab=0 ⇔ a=0 or b=0
An ED
A non-zero ring R is a Euclidean Domain ED if it is an integral domain equipped with a function V: R{0} → |N0 s.t for a,b in R, with a≠0, there exists q, r in R s.t b=aq+r with ether r=0 or V(r) < V(a)
Gaussian Integers
Z[i] = {a+bi | a,b in Z}
Euclidean Function Norm
N(a) = aa¯ = c2 +d2
a= c+id
gcd in a ring
In a ring R, a gcd of two elements a, b is an element d satisfying:
(i) d|a and d|b
(ii) c|a and c|b ⇒ c|d
Irreducible
In an Integral Domain, an element p is called irreducible if it is neither 0 nor a unit and p=ab ⇒ either a or b is a unit
Prime (In an integral domain)
In an integral domain R, p is called prime if it is neither 0 nor a unit and p|ab ⇒ either p|a or p|b.
A Unique Factorisation Domain UFD
An ID, R, is a UFD if:
- Every non zero element may be expressed as a unit times a product of irreducibles.
- The factorisation in (1) is unique up to factors and replacing the irreducibles by associates.
That is, if a in R is nonzero and
a = u p1p2p3….pr = u q1q2….qs
with units u,v and all pi, qi irreducible then r=s and after permuting the qi as necessary, there are vj units, 1 <= j <= r such that qj=vjpj and u = v v1v2…vr.
