Commutative Algebra

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21 domain-specific abstractions whose origin domain is Commutative Algebra.

• Arf ring — A one-dimensional semilocal Cohen–Macaulay ring satisfying the Arf closure condition that controls integrally closed ideals and multiplicity sequences.
• Artin–Tate lemma — If A is commutative Noetherian, C is a finite-type A-algebra and finite as a module over an intermediate A-subalgebra B, then B is finite type over A.
• Associated graded ring — The graded ring formed from successive quotients of powers in an ideal filtration, preserving leading-order information while discarding higher filtration terms.
• Cohen ring — A field or complete discrete valuation ring of mixed characteristic whose maximal ideal is generated by the residue characteristic, used to lift residue fields in local algebra.
• Commutative ring — A ring whose multiplication is commutative, providing the algebraic setting in which ideals, localization, spectra and polynomial geometry acquire their standard symmetric forms.
• Completion of a ring — Replace a ring by the inverse limit of its quotients by successive powers of an ideal, producing an ideal-adically complete ring together with the canonical map from the original ring.
• Connected ring — A commutative ring with no idempotents other than zero and one, equivalently one whose prime spectrum is connected in the Zariski topology.
• Depth (ring theory) — A homological invariant measuring the length of a maximal regular sequence acting on a module, equivalently the first degree of nonvanishing Ext under standard local Noetherian hypotheses.
• Deviation of a local ring — A sequence of nonnegative homological invariants counting generators in an acyclic closure and measuring successive departures of a local ring from regularity and complete-intersection structure.
• Formal power series — An infinite coefficient sequence manipulated as an algebraic series in an indeterminate, without any requirement that numerical substitution converge.
• Frobenius endomorphism — The natural ring endomorphism x↦xᵖ on a commutative ring of prime characteristic p, becoming an automorphism precisely in important perfect cases.
• Hasse–Schmidt derivation — A sequence of additive maps encoding a formal higher-order derivation through a multiplicative generating-series identity.
• Kähler differential — The universal module-valued derivation that algebraically represents first-order differentiation for a ring map.
• Localization (commutative algebra) — The construction that formally inverts a multiplicative subset of a commutative ring or module, creating fractions that focus algebra on a chosen region or prime.
• Matrix factorization of a polynomial — A pair of square matrices over a polynomial ring whose two products both equal multiplication by a fixed polynomial times the identity.
• Multiplicatively closed set — A subset of a ring containing the multiplicative identity and closed under every finite product.
• Nilradical of a ring — The ideal of all nilpotent elements in a commutative ring, equivalently the radical of the zero ideal and the intersection of all prime ideals.
• Prime ideal — A proper ideal P of a commutative ring such that ab in P implies a in P or b in P, equivalently making the quotient ring an integral domain.
• Ring of mixed characteristic — A characteristic-zero commutative ring with a quotient or residue field of positive characteristic, usually considered locally at a prime p.
• Seminormal ring — A reduced commutative ring in which compatible square and cube roots already come from one element, preventing certain hidden subintegral identifications.
• Total ring of fractions — Localize a commutative ring at all of its non-zero-divisors, embedding it injectively into the largest localization that makes every regular element invertible without forcing zero divisors to invert.