Divisibility in Integral Domains.

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Recall. Let R be a commutative ring with unity and D an integral domain. An integral domain is a commutative ring with a multiplicative identity (1 ≠ 0) with no zero-divisors, that is, ab = 0 ⇒ a = 0 or b = 0.

A ring R is called a principal ideal domain, PID for short, if it is an integral domain such that every ideal is principal, that is, has the form ⟨a⟩ = {ra | r ∈ R}, e.g., ℤ, F (the only ideals are the trivial ideal ⟨0⟩ and the whole field ⟨1⟩ = F) and F[x] (where F is a field).

Definitions. Suppose that we have two arbitrary elements from the ring, say a, b ∈ R.

• We say that a divides b, and write a|b, if ∃c ∈ R with b = ac.
• We say that a and be are associates if a = ub where u is a unit of D. In the ring of integers ℤ, the units are precisely the integers 1 and -1. If a, b are associates, then a = 1·b = b or a = (-1)·b = -b. Thus, two integers are associates in ℤ if and only if the’re the same up to sign.

  A unit of a ring, u ∈ R, is an invertible element for the multiplication operation of the ring, that is, ∃v ∈ R, vu = uv = 1.

• A non zero, non unit element “a” of an integral domain D is irreducible if when a = bc, then b or c is a unit. In other words, it cannot be written as a non trivial (except by written it as a product of units) product of two elements of the ring.
• A non zero, non unit element a of an integral domain D is prime if when a|bc implies a|b or a|c, that is, if it divides a product, then it divides (at least) one of the terms of the product.

  The concepts of irreducibles and primes are equivalent in the case of integers, but in general they are not.

Examples

• R = ⟨x², y², xy⟩ is a subring of ℚ[x, y]. x², y², and xy are irreducible within R, but xy is not prime because xy | x²y² and yet xy ɫ x² and xy ɫ y²

• Consider the ring $ℤ[\sqrt{-5}]$ = {a + b$\sqrt{-5}$ | a, b ∈ ℤ} and a function N, called the norm, N: $ℤ[\sqrt{-5}]$ → ℕ, N(a + b$\sqrt{-5}$) = a² + 5b² with the following properties:

1. N(x) = 0 ↭ x = 0.
2. N(xy) = N(x)N(y) ∀x, y.
3. u is a unit ↭ N(u) = 1.
4. If N(x) is prime, then x is irreducible in $ℤ[\sqrt{-5}]$

Proof.

• (1⇒) N(x) = 0 ⇒ a² + 5b² = 0 ⇒ [a² ≥ 0, b² ≥ 0] a² = b² = 0 ⇒ a = b = 0 ⇒ x = 0.
• (2) x = a + b$\sqrt{-5}$, y = c + d$\sqrt{-5}$ ⇒ xy = (ac -5bd) + (ad + bc)$\sqrt{-5}$ ⇒ N(xy)= (ac -5bd)² + 5(ad + bc)² = a²c² + 5²b²d² -10acbd + 5a²d² + 5b²c² + 10adbc= a²(c² +5d²) + b²(5c² + 25d²) = a²(c² +5d²) + 5b²(c² + 5d²) = (a² + 5b²)(c² + 5d²) = N(x)N(y)
• (3⇒) Suppose u is a unit ⇒ ∃u^-1 ∈ $ℤ[\sqrt{-5}]$ : u·u^-1 = u^-1u = 1 ⇒ N(u·u^-1) = N(1) = 1 ⇒ [N(xy) = N(x)N(y) ∀x, y] N(u)N(u^-1) = 1 but all of these terms are naturals ⇒ N(u) = 1.
• (3⇐) Suppose u = a + b$\sqrt{-5}$ and N(u) = a² + 5b² = 1 ⇒ b = 0, a = ± 1 ⇒ u = ± 1 and those are both units.
• (4) N(x) is prime, suppose x = yz. N(x) = [N(xy) = N(x)N(y) ∀x, y] N(y)N(z) and N(x) is prime in ℕ ⇒ Without losing generosity (you may need to swap or rename y and z), N(y) = 1 and N(z) = N(x) ⇒ [By 3] y is a unit, and therefore x is irreducible.

The concepts of irreducibles and primes are equivalent in the case of integers, but in general they are not. 2 + $\sqrt{-5}$ is irreducible but not a prime in $ℤ[\sqrt{-5}]$

1. 2 + $\sqrt{-5}$ is irreducible. Suppose 2 + $\sqrt{-5} = xy$ ⇒ N(x)N(y) = N(xy) = N(2 + $\sqrt{-5})$ =[N(x) = a² + 5b²]) = 2² + 5·1² = 9. If x or y is a unit, then we are done∎. Let’s suppose that x and y are not units ⇒ their norms are not equal to one ⇒[N(x)N(y) = 9, N(x), N(y) ∈ ℕ] N(x) = N(y) = 3, but this is impossible because N(x) = a² + 5b² = 3, and a fast inspection yields that there are no integers a, b satisfying this equality ⊥
2. We claim that 2 + $\sqrt{-5}$ is not prime. (2 + $\sqrt{-5})(2 - \sqrt{-5})$ =[(a-b)(a+b) = a²-b²] 4 + 5 = 9 ⇒ $2 + \sqrt{-5}~|~3·3$, but $2 + \sqrt{5}~ ɫ~ 3$

It follows that the factorization of the element 9 into irreducible elements are not unique, 9 = 3·3 = (2 + $\sqrt{-5})(2 - \sqrt{-5})$ ⇒ The ring $ℤ[\sqrt{-5}]$ is not a unique factorization domain, that is, an integral domain in which every nonzero non-invertible element has a unique factorization.

• Consider the ring $ℤ[i\sqrt{3}]$ = {a + bi$\sqrt{3}$ | a, b ∈ ℤ} and a function N, called the norm, N: $ℤ[i\sqrt{3}]$ → ℕ, N(a + bi$\sqrt{3}$) = a² + 3b² with the following properties:

1. N(x) = 0 ↭ x = 0.
2. N(xy) = N(x)N(y) ∀x, y.
3. u is a unit ↭ N(u) = 1.
4. If N(x) is prime, then x is irreducible in $ℤ[i\sqrt{3}]$

Proof.

• (1⇒) N(x) = 0 ⇒ a² + 3b² = 0 ⇒ [a² ≥ 0, b² ≥ 0] a² = b² = 0 ⇒ a = b = 0 ⇒ x = 0.
• (2) x = a + bi$\sqrt{3}$, y = c + di$\sqrt{3}$ ⇒ xy = (ac -3bd) + (ad + bc)i$\sqrt{3}$ ⇒ N(xy)= (ac -3bd)² + 3(ad + bc)² = […] = (a² + 3b²)(c² + 3d²) = N(x)N(y)
• (3⇒) Suppose u is a unit ⇒ ∃u^-1 ∈ $ℤ[i\sqrt{3}]$ : u·u^-1 = u^-1u = 1 ⇒ N(u·u^-1) = N(1) = 1 ⇒ [N(xy) = N(x)N(y) ∀x, y] N(u)N(u^-1) = 1 but all of these terms are naturals ⇒ N(u) = 1.
• (3⇐) Suppose u = a + bi$\sqrt{3}$ and N(u) = a² + 3b² = 1 ⇒ b = 0, a = ± 1 ⇒ u = ± 1 and those are both units.
• (4) N(x) is prime, suppose x = yz. N(x) = [N(xy) = N(x)N(y) ∀x, y] N(y)N(z) and N(x) is prime in ℕ ⇒ Without losing generosity (you may need to swap or rename y and z), N(y) = 1 and N(z) = N(x) ⇒ [By 3] y is a unit, and therefore x is irreducible.

The concepts of irreducibles and primes are equivalent in the case of integers, but in general they are not. 1 + i$\sqrt{3}$ is irreducible but not a prime in $ℤ[i\sqrt{3}]$

1. 1 + i$\sqrt{3}$ is irreducible. Suppose 1 + i$\sqrt{3} = xy$ ⇒ N(x)N(y) = N(xy) = N(1 + i$\sqrt{3})$ =[N(x) = a² + 3b²]) = 4. If x or y is a unit, then we are done∎. Let’s suppose that x and y are not units ⇒ their norms are not equal to one ⇒[N(x)N(y) = 4, N(x), N(y) ∈ ℕ] N(x) = N(y) = 2, but this is impossible because N(x) = a² + 3b² = 2 ⇒ b = 0, a²=2 where a ∈ ℤ ⊥
2. We claim that 1 + i$\sqrt{3}$ is not prime. (1 + i$\sqrt{3})(1 - i\sqrt{3})$ =[(a-b)(a+b) = a²-b²] 4 = 2·2 ⇒ $1 + i\sqrt{3}~|~2·2$, but $1 + i\sqrt{3}~ ɫ~ 2$

Theorem. In an integral domain, every prime element is irreducible.

Proof.

Suppose that a is a prime in an integral domain D and a = bc. Is a irreducible? ↭ Is b or c a unit?

a is prime, a = bc ⇒ a | b or a | c. Let’s assume without any loss of generality a | b ⇒ ∃t ∈ D such that at = b.

An integral domain is a commutative ring with a multiplicative identity with no zero-divisors, therefore 1 ∈ D ⇒ b·1 = b = at = (bc)t =[Associativity] b(ct) ⇒ [In an integral domain, the cancellation property holds] 1 = ct ⇒ c is a unit ∎

Theorem. Let R be a ring which is also a principal ideal domain. Let a ≠ 0 be an element of R. Then, a is irreducible (it cannot be written as a non trivial product -except units- of two elements of the ring) if and only if a is prime. Hence, 💡primes and irreducible elements in a PID are the same.

Proof.

⇐) A ring R is a Principal ideal domain, PID for short, if it is an integral domain such that every ideal is principal ⇒ R is an integral domain ⇒[Theorem. In an integral domain, every prime element is irreducible.] In an integral domain, every prime is irreducible and we are done.

⇒) Let a be an irreducible element of a principal ideal domain. We claim that a is prime.

Let’s suppose that a | bc, a | b or a | c?

We are going to consider the ideal I defined as I = {ax + by | x, y ∈ D} ⇒ [D is a PID, therefore every ideal has the form ⟨a⟩ for some a ∈ D] I = {ax + by | x, y ∈ D} = ⟨d⟩.

a ∈ I = ⟨d⟩ ⇒ a = dr ⇒ [By assumption, a is irreducible] d or r is a unit, so we are faced with just two options,

1. Let’s assume that d is a unit, then I = {ax + by | x, y ∈ D} = ⟨d⟩ =[d is a unit, ∃d^-1∈ D, d·d^-1 = 1 ∈ ⟨d⟩ and consequently I = ⟨d⟩ = D] D ⇒ ∃x, y ∈ D: ax + by = 1 ⇒ c = cax + bcy ⇒ [a | cax, a | bcy (By assumption, a | bc)] a | c∎
2. Let’s assume that r is a unit, then I = {ax + by | x, y ∈ D} = ⟨d⟩ =[a = dr ⇒ ⟨a⟩⊆⟨d⟩. Besides, r is a unit, ∃r^-1∈ D, d = ar^-1 ∈ ⟨a⟩ ⇒ ⟨d⟩⊆⟨a⟩] ⟨a⟩, b ∈ I = ⟨a⟩ ⇒ a | b∎

Proposition. ℤ is a principal ideal domain.

Proof.

Let I be an ideal. If I is the trivial ideal, I = {0}, then I = ⟨0⟩. Let us assume that this is not the case, I ≠ {0}, so there is a smallest positive number on I, say n. We claim that I = ⟨n⟩, obviously ⟨n⟩ ⊆ I.

∀m ∈ I, by the Euclid’s division theorem, m = qn + r, 0 ≤ r < n ⇒ r = m - qn ∈ I ⇒ [By assumption n is the smallest positive number on I, 0 ≤ r < n] r = 0 ⇒ m = qn ⇒ m ∈ ⟨n⟩∎

Proposition. ℤ and F[x] where F is a field are principal ideal domains, but ℤ[x] is not a principal ideal domain.

Proof.

The ideal I = {f(x) ∈ ℤ[x] | f(0) is even} is not principal, that is, of the form ⟨h(x)⟩

By reduction to the absurd, I = {f(x) ∈ ℤ[x] | f(0) is even} = ⟨h(x)⟩

x, 2 ∈ I ⇒ ∃f(x), g(x)∈ ℤ[x] such that 2 = h(x)f(x) and x = h(x)g(x) ⇒ 0 = deg(2) = deg(h(x)) + deg(f(x)) ⇒ deg(h(x)) = deg(f(x)) = 0 ⇒ h(x) is a constant polynomial.

2 = h(x)f(x) ⇒ 2 = h(1)f(1) ⇒ h(1) = ±1 or ±2. Since the constant polynomial 1 is not in I, h(1) = ±2 ⇒ x = h(x)g(x) =[h(x) is a constant polynomial, h(1) = ±2] ±2g(x) for some g(x) ∈ ℤ[x] ⊥

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