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.

There is a rung above this one. universal-algebra (added 2026-08-22 from bergman-invitation-general-algebra) makes the same generalising move a second time: where abstract algebra proves a theorem once for all groups instead of once each for integers, polynomials and permutations, general algebra proves it once for all varieties — every set carrying operations of a given signature subject to given identities — and reads groups, rings and lattices off as instances.

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 e with e∘a = a∘e = a;
an inverse a⁻¹ for each a, with a∘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. baker-algebra-number-theory reverses the order: arithmetic first and ring-free, groups second. See number-theory.

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.

A third treatment, and it fills part of that gap (2026-09-10)

Part IV of gallier-quaintance-math-deep is algebra taught for use downstream rather than as a first course: polynomials and PIDs, annihilating polynomials and primary decomposition, UFDs and Noetherian rings with Hilbert's basis theorem, tensor and exterior algebras, modules over a PID, and normal forms up to the rational canonical form. It is the corpus's first source on modules and tensor algebra, and it reaches them from linear algebra rather than from group theory — the canonical-forms chapter is where the two halves meet.

What the corpus still lacks

Representation theory appears in none of the three books. 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.

judson-abstract-algebra · lal-algebra-1 · gallier-quaintance-math-deep · group-theory · linear-algebra · mathematics · real-analysis · synthesis