A local ring is a commutative ring with exactly one maximal ideal. The elements outside this maximal ideal are precisely the units. Local rings arise naturally from localizing any ring at a prime ideal, and they represent the algebraic analog of "zooming in on a single point" in geometry. Working locally simplifies many problems because the ideal structure collapses to a single chain above the maximal ideal.
A local ring is a commutative ring with exactly one maximal ideal, traditionally denoted (R, 𝔪). Equivalently, the set of non-units forms an ideal — which is then automatically the unique maximal ideal. The quotient k = R/𝔪 is a field called the residue field. Local rings are the algebraic structures that describe "what happens at a single point," and most of commutative algebra operates by reducing questions to the local case.
The most important source of local rings is localization at a prime ideal. If 𝔭 is a prime ideal of R, then R_𝔭 = S⁻¹R (where S = R \ 𝔭) is a local ring with maximal ideal 𝔭R_𝔭. For example, ℤ₍₅₎ consists of fractions a/b where 5 does not divide b, and its unique maximal ideal is 5ℤ₍₅₎. In this ring, 2, 3, 7, and all primes other than 5 become units (they are invertible), and the only "interesting" arithmetic is divisibility by 5. The residue field is ℤ₍₅₎/5ℤ₍₅₎ ≅ 𝔽₅.
Local rings also arise as quotients and completions. The ring k[x]/(x²) — the "dual numbers" over k — is local with maximal ideal (x̄). It has a single "infinitesimal direction" represented by x̄, with x̄² = 0. In algebraic geometry, this ring describes the first-order neighborhood of a point, and maps from Spec(k[x]/(x²)) into a variety represent tangent vectors. The power series ring k[[x]] is a complete local ring with maximal ideal (x), modeling the "formal" neighborhood of a point.
The power of local rings comes from the local-global principle: many module-theoretic properties (being zero, being free, being finitely generated) can be checked locally — that is, after localizing at every maximal ideal. Since localizations at maximal ideals are local rings, this reduces questions about general rings to questions about local rings. In the local setting, you have tools like Nakayama's lemma, the structure theory of regular local rings, and completion, none of which are available globally. This is why the passage from global to local is the most common first move in commutative algebra.