87 articles
Hilbert's Nullstellensatz
Hilbert's Nullstellensatz, German for "zero locus theorem," is David Hilbert's 1893 theorem bridging commutative algebra and algebraic geometry: over an algebraically closed field, ideals of polynomials correspond precisely to the sets of their common zeros.
Homological conjectures in commutative algebra
The homological conjectures are a web of related conjectures about homological properties of commutative rings, collected by Melvin Hochster in 1975; since 2016 perfectoid geometry has proved the direct summand conjecture.
I-adic completion
The I-adic completion of a ring R at an ideal I is the inverse limit of the quotients R/Iⁿ, replacing polynomials by formal power series and yielding the p-adic integers.
Ideal (ring theory)
An ideal in ring theory is a subset of a ring closed under addition and multiplication by any ring element, introduced by Richard Dedekind in 1871.
Idempotence
Idempotence is the property of an element e of a ring satisfying e² = e, equivalently e(1 − e) = 0; multiplication by e acts like a projection.
Injective module
An injective module is a module J over a ring R such that every homomorphism from a submodule into J extends to the whole module, the dual of projectivity.
Integral domain
An integral domain, also called an entire ring, is a commutative ring in which ab = 0 forces a = 0 or b = 0; the integers are the standard example.
Integral element
An integral element of a ring S over a base ring R satisfies a monic polynomial with coefficients in R, generalizing algebraic elements of field extensions.
Integrally closed domain
An integrally closed domain is an integral domain containing every element of its fraction field integral over it, that is, satisfying a monic polynomial over the domain.
Irreducible element
An irreducible element, also called an atom, is a nonzero, non-unit element of an integral domain that cannot be written as a product of two non-unit elements; it generalizes prime numbers to abstract rings.
Irreducible polynomial
An irreducible polynomial is a nonconstant polynomial that cannot be written as a product of two lower-degree polynomials, the polynomial analog of a prime number, with irreducibility depending on the coefficient field.
Jacobson radical
The Jacobson radical of a ring R, written J(R), is the intersection of all maximal left ideals, equivalently of all maximal right ideals; it was introduced by Nathan Jacobson in 1945.
Jacobson ring
A Jacobson ring, also called a Hilbert ring, is a commutative ring in which every radical ideal is the intersection of the maximal ideals containing it.
Krull dimension
The Krull dimension of a commutative ring is the supremum of the lengths of chains of prime ideals, defined by Wolfgang Krull in 1928 as the standard notion of dimension in commutative algebra.
List of number fields with class number one
The list of number fields with class number one covers rings of integers with unique factorization, including the nine imaginary quadratic discriminants 3, 4, 7, 8, 11, 19, 43, 67, and 163.
Local ring
A local ring, also known as a quasi-local or quasilocal ring, is a ring with exactly one maximal ideal, the basic algebraic tool for studying geometric objects near a single point.
Localization (ring theory)
Localization is a construction in commutative algebra that inverts a multiplicative set S in a ring, producing fractions S⁻¹A; at a prime ideal it yields a local ring.
Maximal ideal
A maximal ideal is a proper ideal of a ring with no ideal strictly between it and the ring; an ideal is maximal exactly when the quotient ring is a field.
Monic polynomial
A monic polynomial is a nonzero polynomial whose leading coefficient is exactly 1. Products of monic polynomials stay monic, and division by a monic works over any ring.
Near-ring
A near-ring is an algebraic structure with addition and multiplication in which addition forms a possibly nonabelian group and multiplication distributes over addition from one side only.
Nilpotent element
A nilpotent element of a ring is an element x whose some power x raised to n equals 0, making every nilpotent a zero divisor though not conversely, as with 2 in the integers mod 6.
Nilradical
The nilradical of a commutative ring is the ideal of all nilpotent elements, also equal to the intersection of all prime ideals of the ring.
Noether normalization lemma
The Noether normalization lemma, a result in commutative algebra proved by Emmy Noether in 1926, states that every finitely generated algebra over a field is a finite extension of a polynomial subring.
Noetherian ring
A Noetherian ring is a ring in which every ideal can be generated by finitely many elements, named after Emmy Noether, who studied them in 1921.
Non-unique factorization
Non-unique factorization is the phenomenon in rings of algebraic integers and factorization monoids where an element has essentially different irreducible factorizations, classically 6 = 2·3 = (1+√−5)(1−√−5) in Z[√−5], measured by invariants like elasticity.
Order (ring theory)
In ring theory, an order is a subring of the ring of integers of a number field that has finite index there, a concept introduced by Richard Dedekind.
Polynomial ring
A polynomial ring, also called a polynomial algebra, is a ring whose elements are polynomials in one or more indeterminates, written R[x] over a base ring R.
Primary decomposition
Primary decomposition represents an ideal of a commutative ring as an intersection of primary ideals, generalizing prime factorization of integers; it was proved for Noetherian rings by Emmy Noether.
Primary ideal
A primary ideal is a proper ideal of a commutative ring such that a product in it forces one factor, or a power of the other, into it.
Prime and irreducible elements
Prime and irreducible elements are related notions in ring theory: every prime element of an integral domain is irreducible, while the converse fails, as in Z[√−5].