Rings and Fields

From Groups to Rings

The integers ℤ have two operations: addition and multiplication. The generalization is a ring.

A ring (R, +, ·): (R, +) is an abelian group, multiplication is associative, distributivity holds: $a(b+c) = ab+ac$ and $(a+b)c = ac+bc$.

A ring is commutative if $ab = ba$. It is unital if there is a 1.

Examples: ℤ, ℝ[x] (polynomials), M(n,ℝ) (square matrices), ℤ/nℤ.

Ideals and Quotient Rings

An ideal $I \subseteq R$ is a subgroup under addition, closed under multiplication by ring elements: $r \cdot I \subseteq I$ and $I \cdot r \subseteq I$.

The quotient ring $R/I$: elements are cosets $a + I$. This is a ring with operations $(a+I) + (b+I) = (a+b)+I$, $(a+I)(b+I) = ab+I$.

Homomorphism Theorem: If $\varphi: R \to S$ is a ring homomorphism, then $R/\ker \varphi \cong \operatorname{Im} \varphi$.

In ℤ: $nℤ = {0, \pm n, \pm 2n, \ldots}$ is an ideal. ℤ/nℤ is the ring of residues.

In ℝ[x]: $(x^2+1)$ is an ideal. $\mathbb{R}[x]/(x^2+1) \cong \mathbb{C}$ — this is the construction of complex numbers!

Prime and Maximal Ideals

An ideal $P$ is prime if $ab \in P \implies a \in P$ or $b \in P$. Maximal if $P \ne R$ and there is no ideal strictly between $P$ and $R$.

In a commutative ring: $I$ is maximal $\iff R/I$ is a field. $I$ is prime $\iff R/I$ is an integral domain.

Fields

A field is a commutative ring with unity in which every nonzero element is invertible.

Examples: ℚ, ℝ, ℂ, ℤ/pℤ (p is prime), ℚ($\sqrt{2}$) = ${a + b\sqrt{2} : a, b \in ℚ}$.

Characteristic of a field: the smallest $p$ such that $1+1+\ldots+1$ ($p$ times) $= 0$. If there is none — char $= 0$ (ℚ, ℝ, ℂ). Finite fields have characteristic $p$ (prime).

Finite Fields GF($p^n$)

For every prime $p$ and natural $n$, there exists a unique (up to isomorphism) field $\mathrm{GF}(p^n)$ of order $p^n$. These fields are fundamental in coding theory and cryptography.