Divisibility in ℤ
A Fundamental concept in ℤ is divisibility.
Defintion 2.2.1. Let a,b in ℤ. We say that b divides a (or that b is a divisor of a), and we write b|a, if a = bc for some c ∈ ℤ. In this case we say also that the integer a is a multiple of the integer b. If b does not divide a, we write b ∤ a.
For example 3|12, 4|−16 by 4 ∤ 2 e 0 ∤ a if a ≠ 0.
A ring R is said to contain zero-divisors if it contains one or more nonzero elements that when multiplied together give the result 0.
Definition 2.2.2. In a commutative ring, an element a ≠ 0 is said zero divisor if there exists b ≠ 0 such that ab = 0. □
Definition 2.2.3. An integral domain is a commutative ring without zero divisors. □
Proposition 2.1.2 says that the ring ℤ is an integral domain.
Definition 2.2.4. It is said common divisor of elements a and b in ℤ an element c ∈ ℤ such that c|a and c|b. □
Lemma 2.2.5. If c is a common divisor of a and b, then c divides any integer of the form sa + tb, with s and t in ℤ, in formula:
c|a and c|b ⇒ c|sa + tb ∀s,t ∈ ℤ
Proof. c|a ⇒ a = ch for some h ∈ ℤ; c|b ⇒ b = ck for some k ∈ ℤ. Thus, for each s, t in ℤ, sa + tb = s(ch) + t(ck) = c(sh + tk), consequently c|sa + tb.□
Corollary 2.2.6 - If c|a and c|a + b then c|b
Proof. If c is a common divisor of a and a + b, then for s = −1 and t = 1 we get that it divides (−1) ⋅ a + 1 ⋅ (a + b) = b. □
We introduce now some definitions important in the study of rings, that we shall use throughout the course.
Definition 2.2.7. An element u ∈ ℤ which divides 1 is called unit (or invertible element) of ℤ. □
It is immediate to find that the only units of ℤ are 1 and −1.
Definition 2.2.8. Two elemens a and b of ℤ such that a|b and b|a are said associates. □
It follows immediately from the definition that two elements a and b are associates if and only if a = bu, with u is a unit. Thus, in ℤ two elements are associates if and only if they differ by the sign. The relation of being "associates" is an equivalence relation.
Definition 2.2.9. An element a ∈ ℤ which is nonzero, not a unit is irreducibile, if a = bc, with b,c in ℤ implies that either b or c is a unit. □
Definition 2.2.10. A non-zero element p ∈ ℤ which is not a unit, is said prime, if when p|ab with a,b in ℤ, p divides at least one of the two factors a or b. □
Remarks
The definition just given of irreducible element corresponds to the common definition of prime number in ℤ, i.e.:
Proposition 2.2.11. An integer p > 1 is called a prime number if its only divisors are 1 and p. An integer a > 1 is called composite if it is not prime.
However in the previous treatement we were talking about irreducible elements and not of prime numbers. We'll see that the two notions of prime element and irreducibile element are the same in ℤ, even though this is not the general case. At this point we are able to prove that a prime element is necessarily irreducibile.
Example. For example if n = 6, a = 3, b = 2, n|ab, but n is not prime and doesn't divide neither a nor b.
One of the very first observations to be made from this definition is that being prime is a stronger condition than being irreducible:
Proposition 2.2.12. In any integral domain (e.g., ℤ), any prime element is irreducible.
Proof. Let r be a prime element of the integral domain D. Suppose, by way of contradiction that r is not irreducible, so that r = ab, where a and b are non-units. Then r divides ab, and so, as r is prime, r divides one of the two factors a or b — say a. Then rh = a for some h ∈ D and r = rhb . But then, by the Cancellation laws, 1 = hb, whence b is a unit, contrary to our assumption. Thus r is irreducible. □
In general, amon integral domains, an irreducible element may not be prime. However it is true for principle ideal domains. We'll show that irreducible elements in ℤ are also prime later in Proposition 2.3.8.