Abstract algebra
The study of algebraic structures — a set together with operations satisfying stated axioms — rather than of the numbers those operations were first defined on. Prove a theorem once about every structure with a given signature, and it holds for integers, polynomials, matrices, permutations and symmetries at no extra cost.
Two texts here: lal-algebra-1 (Springer, paid) and judson-abstract-algebra (GFDL, free). The definitions below are read from Judson.
The structures, and how they nest
A group is a set with one associative operation, an identity, and inverses:
- associativity —
(a∘b)∘c = a∘(b∘c); - an identity
ewithe∘a = a∘e = a; - an inverse
a⁻¹for eacha, witha∘a⁻¹ = a⁻¹∘a = e.
Add a∘b = b∘a and the group is abelian. The integers under addition are the standard example; the
symmetries of an equilateral triangle, written as a Cayley table, are the standard non-example of
commutativity and the reason groups are the algebra of symmetry (group-theory).
A ring adds a second operation. Judson’s framing is that a ring is an abelian group under addition together with a multiplication satisfying the remaining axioms — so addition is fully invertible and multiplication is not. From there the hierarchy is a sequence of extra demands:
| Structure | What is added |
|---|---|
| Ring | two operations; (R,+) abelian |
| Ring with unity | a 1 ≠ 0 acting as multiplicative identity |
| Commutative ring | ab = ba |
| Integral domain | commutative, with unity, and no zero divisors: ab = 0 forces a = 0 or b = 0 |
| Division ring | unity, and every non-zero element has an inverse |
| Field | a commutative division ring |
The examples do the work of the definitions. ℤ is an integral domain but not a field — 2 has no
integer inverse. ℚ, ℝ and ℂ are fields. ℤₙ is a commutative ring that may fail to be a domain:
in ℤ₁₂, 3·4 ≡ 0, so 3 and 4 are zero divisors. 2×2 real matrices form a noncommutative ring
where AB = 0 is possible with neither factor zero. The quaternions are a noncommutative division
ring — everything invertible, nothing commuting.
Why the rest of the corpus needs it
linear-algebra is the visible dependency: a vector space is defined over a field, so every theorem the corpus already holds about vector spaces is a theorem about a structure whose scalars satisfy the field axioms. Judson makes the dependency explicit by putting vector spaces in chapter 20, after rings and immediately before field extensions.
Beyond that: modular arithmetic and public-key cryptography are group and ring theory (judson-abstract-algebra chapter 7), error-correcting codes are linear algebra over finite fields (chapter 8, and again in chapter 22), and Boolean algebra formalises circuits (chapter 19). The corpus’s open information and coding theory edge runs straight through this subject.
Number theory is entangled, not adjacent
Both texts return to arithmetic after introducing rings, on the argument that the ring is the right language for statements about integers. lal-algebra-1 develops the number systems from Peano’s axioms, does linear Diophantine equations and congruences early, then comes back for arithmetic functions, quadratic residues, Gauss’s quadratic reciprocity law and nth-power residues, and finally “Arithmetic in Rings.” Judson reaches Fermat’s and Euler’s theorems as corollaries of Lagrange’s.
Two treatments, one subject
They disagree about what a first course is for, which is more useful than agreement would be:
- Foundations first vs proofs as needed. Lal spends two chapters on logic (Russell’s and Cantor’s paradoxes as the motivation) and set theory (Zermelo–Fraenkel plus choice) before any algebra. Judson opens with a short note on proofs and gets to groups in chapter 3.
- Structure theory vs applications. Lal goes deep into classification — Sylow, finite abelian groups, Schreier and Jordan–Hölder, Remak–Krull–Schmidt, solvable and nilpotent groups, presentations. Judson spends the equivalent space on cryptography, coding theory and Galois theory.
- Reach. Judson finishes at the fundamental theorem of Galois theory in one volume; Lal is volume one of three and defers it.
What the corpus still lacks
Modules and representation theory appear in neither book. Category-theoretic framing appears in neither. And this page rests on one book read plus one book’s chapter abstracts — lal-algebra-1‘s body has not been read here, so where the two are described as differing, that is a difference in stated scope, not a comparison of exposition.
Related
judson-abstract-algebra · lal-algebra-1 · group-theory · linear-algebra · mathematics · real-analysis · synthesis