Rings keep addition fair and let multiplication misbehave
A ring has two operations modeled on the integers: addition forms an abelian group, multiplication is associative and distributive, and (in the modern default) a multiplicative identity exists. Multiplication need not commute—and that is exactly where matrices shine.
Elements may be numbers, polynomials, square matrices, functions, or power series. Formally the additive structure is an abelian group; multiplication distributes on both sides and has a unit 1 in the terminology used here. Authors who drop the unit call the weaker structure a rng—"ring" missing an "i"—and the even integers are a classic rng that is not a ring. Commutative rings, where ab = ba, support commutative algebra, deeply tied to algebraic number theory and algebraic geometry. Fields are commutative division rings; division rings need inverses for every nonzero element without requiring commutativity.
Examples clarify the split. Fields, the integers, polynomial rings, coordinate rings of affine varieties, and rings of integers of number fields are commutative. Noncommutative examples include n × n real matrices for n ≥ 2, group rings, operator algebras, rings of differential operators, and cohomology rings. Conceptualization stretched from the 1870s to the 1920s with Dedekind, Hilbert, Fraenkel, and Noether, generalizing Dedekind domains and polynomial and invariant rings before spreading into geometry and analysis.
Axioms list associativity and commutativity of addition, an additive identity and inverses, associativity of multiplication, a multiplicative identity, and both distributive laws. Books in commutative algebra often quietly redefine "ring" to mean commutative ring. A hierarchy climbs from rngs through domains and unique-factorization settings up to fields. Wherever you need integer-like arithmetic without forcing every nonzero element to invert—or without forcing ab = ba—you are in ring territory.
Source: Ring (mathematics)
