10  Rings and Fields

Groups have one operation. But \(\mathbb{Z}\), \(\mathbb{Q}\), \(\mathbb{R}\), and \(\mathbb{C}\) — the number systems Number Systems built — all carry two, addition and multiplication, and the two don’t sit side by side unrelated: \(3 \times (4 + 5) = 3\times4 + 3\times5\) is doing real work every time you expand an expression. Naming what it takes for two operations to interact usefully is this chapter’s whole job, and it finally makes precise a word Vector Spaces already leaned on without defining: field.

10.1 The problem, before the definition

\(\mathbb{Z}\) under addition alone is a group — Groups already covered that. But \(\mathbb{Z}\) also has multiplication, and the two operations are clearly not independent: distributing multiplication over addition is something you do without thinking, in arithmetic and in algebra alike. A structure with one operation can’t express that interaction at all — describing it needs a set with two operations and a rule connecting them.

Two more examples, chosen because they look nothing like numbers, make the same point Vector Spaces made with arrows and polynomials: the set of \(n \times n\) matrices has both addition and multiplication, distributing over each other exactly the way numbers do — even though matrix multiplication isn’t commutative (\(AB \neq BA\), in general). And polynomials, already met as vectors under addition and scaling, also multiply: \((x+1)(x-1) = x^2 - 1\), with the same distributive behavior.

10.2 The abstraction

A ring is a set \(R\) with two operations, \(+\) and \(\times\), such that:

1. \((R, +)\) is an abelian group (see Groups) — this alone is everything addition needs to behave like addition.
2. \(\times\) is associative.
3. Distributivity: \(a(b+c) = ab+ac\) and \((a+b)c = ac+bc\) for all \(a,b,c \in R\).

That’s the core of it — conventions vary on whether a ring is required to have a multiplicative identity \(1\) (this site assumes it does, since every ring below has one) or whether \(\times\) must be commutative (it need not be: matrices are the standing counterexample). A field is a ring with two extra requirements: \(\times\) is commutative, and every nonzero element has a multiplicative inverse — equivalently, \((R \setminus \{0\}, \times)\) is itself an abelian group. \(\mathbb{Q}\), \(\mathbb{R}\), and \(\mathbb{C}\) are fields; \(\mathbb{Z}\) is a ring but not a field, since only \(1\) and \(-1\) have multiplicative inverses in \(\mathbb{Z}\).

This is the word Vector Spaces used before defining: a vector space is built “over a field” precisely because scalar multiplication needs to be undoable — solving \(c\vec{v} = \vec{w}\) for a scalar \(c\) requires dividing, which requires the scalars to have multiplicative inverses. A ring without them isn’t rich enough to serve as a vector space’s scalars in the usual sense.

10.3 Why distributivity, specifically

Closure, associativity, identity, and inverses on \(+\) are just the group axioms again, already justified in Groups. Distributivity is the one genuinely new axiom here, and it’s worth seeing what fails without it: with two operations that satisfy their own axioms but don’t distribute over each other, \(a(b+c)\) and \(ab+ac\) would be two unrelated quantities with no guaranteed relationship — every algebraic manipulation that expands or factors an expression would become invalid, since expanding is an appeal to distributivity. A ring without distributivity isn’t a weaker ring; it’s just two unconnected groups sharing a set, which is a much less useful thing to have a name for.

10.4 What follows, and what a ring is NOT

Every field is a ring, but the reverse is false — \(\mathbb{Z}\) is the standing example, and it’s worth being specific about what it lacks: not associativity or distributivity (it has both), but multiplicative inverses. A ring can also fail to be a field in a second, more surprising way that \(\mathbb{Z}\) doesn’t illustrate: it can have zero divisors — nonzero elements \(a, b\) with \(ab = 0\). This can’t happen in a field (if \(ab=0\) and \(a \neq 0\), multiplying by \(a^{-1}\) forces \(b=0\)), but it happens routinely in the ring of \(n \times n\) matrices, where two nonzero matrices can multiply to the zero matrix. Zero divisors are exactly what block a ring from being a field even when every element is “nonzero” in the usual sense — another instance of this site’s recurring lesson that dropping one axiom can break a consequence you might not have expected it to.

A ring homomorphism is, unsurprisingly by now, a map respecting both operations — the same idea What Is a Mathematical Structure? introduced for groups, applied to a richer structure.

10.5 A word with a history

“Ring” (German Ring, or the fuller Zahlring, “number ring”) is usually credited to David Hilbert’s 1897 report on algebraic number theory, though the underlying idea — sets of algebraic numbers closed under addition and multiplication — traces back further, to Richard Dedekind’s work on ideals in the 1870s–80s. “Field” (German Körper, literally “body”) is Dedekind’s own term, from around 1871, for exactly the structure defined above: a set where all four arithmetic operations (except division by zero) stay inside the set. Both terms predate the fully abstract axiomatic treatment above by decades, in the same historical/formal split the map keeps returning to — mathematicians were studying specific rings and fields of numbers long before “ring” and “field” were given axioms independent of any particular numbers.

10.6 Where this leads

Rings and fields finish the algebraic side of what Groups started, and they settle a debt Vector Spaces left open — “a field” now means something specific. What vector spaces still lack, formally, is any notion of distance or closeness at all, and that’s exactly the gap Metric Spaces fills next.