MAS4002: Algebraic Number Theory Rings and ideals: a summary

In this course all rings will be commutative. A ring is a set R with operations of addition and multiplication such that

• addition and multiplication are commutative: x + y = y + x and xy = yx for all x, y ∈ R;

• addition and multiplication are associative: x + (y + z) = (x + y) + z and x(yz) = (xy)z for all x, y, z ∈ R;

• the distributive law holds x(y + z) = xz + yz for all x, y, z ∈ R;

• R has a zero element 0: 0 + x = x for all x ∈ R;

• R has a unit element 1: 1x = x for all x ∈ R;

• each element of R has a negative: if x ∈ R there is −x ∈ R with x + (−x) = 0.

The standard examples of rings are Z, Q, R and C, the rings of integers, rational numbers, real numbers and complex numbers respectively. Another ring we use frequently is Z_{p}, the ring of integers modulo the prime p. Given a ring R, the polynomial ring R[ X] over R is defined to be the set of all polynomials a_{0} + a_{1}X + a_{2}X^{2} + ⋯ + a_{n}X^{n} with each a_{j} ∈ R, under the operations of polynomial addition and multiplication.

A ring is an integral domain if it has more than one element, and whenever x and y are nonzero elements of R then so is xy. The units of a ring R are the elements a ∈ R which have a multiplicative inverse: i.e., there exists b ∈ R with ab = 1. The set of units of R is denoted by U(R) and forms an abelian group. A ring is a field if it has more than one element, and each nonzero element is a unit. If R is an integral domain and a, b ∈ R we say that a divides b (written a | b) if there is a c ∈ R with b = ac. An element a of an integral domain R is irreducible if a is neither zero nor a unit, and if whenever a = bc with b, c ∈ R then either b or c is a unit.

A subring S of a ring R is a subset of R such that

• S is closed under addition, multiplication and negation: x + y, xy, −x ∈ S for all x, y ∈ S.

• 1 ∈ S.

It follows that S itself is a ring. An ideal I of a ring R is a subset of R such that

• I is non-empty;

• I is closed under addition: a + b ∈ I for all a, b ∈ I;

• xa ∈ I for all x ∈ R and a ∈ I.

A principal ideal of R is one of the form

⟨a⟩ = {xa : x ∈ R}.

More generally a finitely generated ideal is one of the form

⟨a_{1}, a_{2}, …, a_{k}⟩ = {x_{1}a_{1} + x_{2}a_{2} + ⋯ + x_{k}a_{k} : x_{1}, x_{2}, …, x_{k} ∈ R}.

A homomorphism between rings R and S is a function ψ : R → S such that • ψ preserves addition and multiplication: ψ(x + y) = ψ(x) + ψ(y) and ψ(xy) = ψ(x)ψ(y)

for all x, y ∈ R;

• ψ(1) = 1.

A homomorphism is an isomorphism if it is a bijection. In this case its functional inverse is also an isomorphism. Rings R and S are isomorphic (denoted R ≅ S) if there is an isomorphism ψ : R → S. The image of a homomorphism ψ : R → S is its set-theoretic image:

im ψ = {ψ(x) : x ∈ R}

which is a subring of S. The kernel of ψ is

ker ψ = {x ∈ R : ψ(x) = 0};

this is an ideal of R. Given a ring R and ideal I, the quotient ring R/I is the set of cosets x+I for x ∈ R (I is a subgroup of R under addition) with the operations (x+I)+(y+I) = x+y+I and (x + I)(y + I) = xy + I. The projection map π : R → R/I given by π(x) = x + I is a surjective homomorphism from R to R/I. Given any homomorphism then ψ : R → S then

R/ker ψ ≅ im ψ,

the isomorphism given by x+ker ψ ↦ ψ(x). This is the First Isomorphism Theorem for rings. Some important homomorphisms are the evaluation homomorphisms: given α ∈ C we define a homomorphism ϕ_{α} : Q[ X] → C by ϕ_{α}(f) = f(α). Another example is reduction modulo p, ψ : Z → Z_{p} where ψ(a) = \overline{a}. A related example is the homomorphism θ : Z[ X] → Z_{p}[ X] given by

θ(a_{0} + a_{1}X + ⋯ + a_{n}X^{n}) = \overline{a_{0}} + \overline{a_{1}}X + ⋯ + \overline{a_{n}}X^{n}.

An integral domain R is Euclidean if there is a function ϕ : R\{0} → Z with the following properties

• ϕ(a) ≥ 0 for all a ∈ R \ {0};

• ϕ(ab) ≥ ϕ(a) for all a, b ∈ R \ {0};

• if a, b ∈ R and b ≠ 0, then either b | a or there is q ∈ R with ϕ(a − qb) < ϕ(b).

In a Euclidean domain all ideals are principal; in fact if I is a non-zero ideal in the Euclidean domain R, then I = ⟨a⟩ where a is any non-zero element of I minimizing ϕ(a). The standard examples of Euclidean domains are Z and K[ X] where K is any field: the Euclidean functions are given by ϕ(a) = |a| and ϕ(f) = deg f respectively. If a, b ∈ R with R Euclidean, then ⟨a, b⟩ = ⟨g⟩ for some g ∈ R: g is a greatest common divisor of a and b; we have g | a, g | b and there are r, s ∈ R with g = ra + sb. If a is an irreducible element of a Euclidean domain, and a | bc with b, c ∈ R then either a | b or a | c. From this it follows that each Euclidean domain has unique factorization, i.e., if

a_{1}a_{2} ⋯ a_{r} = b_{1}b_{2} ⋯ b_{s}

with the a_{j}s and b_{k}s irreducible elements of R, then r = s and the b_{k}s can be reordered so that a_{j} = u_{j}b_{j} for each j, where u_{j} ∈ U(R).