# All pages

- Adequate domain
- Affine algebraic variety
- Affine domain
- Affine ring
- Affine scheme
- Algebraic extension
- Algebraic norm in a number field
- Annihilator of Noetherian module has Noetherian quotient
- Applying Artin-Rees lemma
- Applying Noetherianness
- Artin-Rees lemma
- Artin-Tate lemma
- Artinian implies Cohen-Macaulay
- Artinian implies IZ
- Artinian ring
- Associate elements
- Associate implies same orbit under multiplication by group of units in integral domain
- Associated prime to a module
- Associated primes turns short exact sequences to sub-unions
- Automorphism-invariant Euclidean norm
- Automorphism-invariant norm
- Automorphism group acts transitively on fibers of spectrum over fixed-point subring
- Bezout domain
- Bezout implies gcd
- Bezout ring
- Birational map of varieties
- Blowup algebra
- Book:Eisenbud
- Cancellative ring
- Cartier divisor
- Catenary module
- Catenary ring
- Cayley-Hamilton theorem
- Characteristic ideal
- Characteristic of a ring
- Characteristic subring
- Characteristically Euclidean domain
- Chinese remainder theorem
- Classification of norm-Euclidean imaginary quadratic integer rings
- Codimension of an ideal
- Coefficient field
- Cohen-Macaulay ideal
- Cohen-Macaulay implies universally catenary
- Cohen-Macaulay is polynomial-closed
- Cohen-Macaulay ring
- Cohen structure theorem
- Coherent ring
- Commutative unital ring
- Compact space
- Complete intersection
- Complete local ring
- Complete system of prime ideals
- Completion of a ring
- Conductor of a numerical semigroup
- Connected scheme
- Content of a polynomial
- Contraction of an ideal
- Convention:Commutative unital rings
- Dedekind-Hasse norm
- Dedekind-Hasse norm implies principal ideal ring
- Dedekind domain
- Dedekind not implies PID
- Depth of an ideal
- Determinantal ideal theorem
- Determinantal ring
- Dimension of an ideal
- Divided polynomial ring
- Dominant rational map
- Effect of ideal contraction on Galois correspondent
- Eisenstein's criterion
- Element of minimum Dedekind-Hasse norm is a unit
- Element of minimum norm among non-units in Euclidean ring is a universal side divisor
- Element of minimum norm in Euclidean ring is a unit
- Elementary divisor domain
- Elementary divisor ring
- Elements in same orbit under multiplication by group of units are associate
- Equicharacteristic ring
- Equidimensional catenary ring
- Equidimensional ring
- Equivalence of definitions of Cohen-Macaulay ring
- Equivalence of dimension notions for Noetherian local ring
- Equivalence of dimension notions for affine domain
- Euclidean domain
- Euclidean implies Dedekind-Hasse
- Euclidean implies principal ideal
- Euclidean norm
- Euclidean not implies norm-Euclidean
- Euclidean ring that is not a field has a universal side divisor
- Euclideanness is localization-closed
- Euclideanness is quotient-closed
- Every Euclidean ring has a unique smallest Euclidean norm
- Every binomial polynomial is irreducible but not prime in the ring of integer-valued polynomials over rational integers
- Every element has power in subring implies bijective on spectra
- Every irreducible is prime implies any two irreducible factorizations are equal upto ordering and associates
- Every proper ideal is contained in a maximal ideal
- Extension of an ideal
- Field
- Filtered ring
- Filtrative Euclidean norm
- Filtrative and multiplicatively monotone Euclidean implies uniquely Euclidean
- Filtrative norm
- Finite-dimensional Noetherian ring
- Finite-dimensional algebra over a field
- Finite-dimensional ring
- Finite morphism
- Finite morphism implies finite on spectra
- Finite ring
- Finitely generated and integral equals finite
- Finitely generated ideal
- Finitely generated module
- Finitely generated morphism
- Formal power series ring
- Free ideal ring
- Free module
- Free resolution
- Fully invariant ideal
- Galois correspondence between a ring and its max-spectrum
- Galois correspondence between a ring and its spectrum
- Galois correspondence induced by a binary relation
- Galois correspondence of extension and contraction
- Galois group
- Galois number field
- Gauss's lemma
- Gcd domain
- Gcd not implies Bezout
- Generalized local ring
- Geometric condition for imaginary quadratic integer ring to be norm-Euclidean
- Global dimension of a ring
- Going-down subring
- Going down
- Going down extension
- Going down for fixed-point subring under finite automorphism group
- Going down for flat extensions
- Going down for integral extensions of normal domains
- Going up extension
- Going up theorem
- Graded Abelian group
- Graded Nakayama's lemma
- Graded module
- Graded ring
- Greatest common divisor
- Grothendieck's generic freeness lemma
- Group of units
- Henselian local ring
- Hereditary ring
- Hilbert-Samuel polynomial
- Hilbert basis theorem
- Hilbert function
- Hilbert polynomial
- Hilbert syzygy theorem
- Homogeneous element
- Homogeneous ideal
- Homomorphism of commutative unital rings
- Ideal
- Ideal-free subring
- Ideal-large subring
- Ideal generated by an irreducible element
- Ideal generated by prime elements
- Ideal generated by two prime elements in a unique factorization domain may be proper and not prime
- Ideal in integral domain implies self-similar
- Ideal of finite colength
- Ideal with maximal radical
- Ideal with prime radical
- Image under leading coefficient map is Noetherian implies finitely generated
- Imaginary quadratic number field
- Indecomposable ideal
- Induced map on spectra by a ring homomorphism
- Injective module
- Injective resolution
- Integral closure of a subring
- Integral domain
- Integral domain in which any two irreducible factorizations are equal upto ordering and associates
- Integral domain in which every irreducible is prime
- Integral domain satisfying ACCP that is not a field has either infinitely many units or infinitely many associate classes of irreducibles
- Integral extension
- Integral extension implies inverse image of max-spectrum is max-spectrum
- Integral extension implies surjective map on spectra
- Integral morphism
- Integrally closed subring
- Intermediate subring condition for ideals
- Interpolation domain
- Intersection of maximal ideals
- Intersection of prime equals radical
- Invertible plus nilpotent implies invertible
- Irreducible element
- Irreducible element not implies prime
- Irreducible element property is not determined by quotient ring
- Irreducible ideal
- Irreducible implies primary (Noetherian)
- Irreducible implies prime (PID)
- Irreducible not implies universal side divisor
- Irreducible ring
- Irreducible scheme
- Isomorphism of varieties
- Jacobson is polynomial-closed
- Jacobson radical
- Jacobson ring
- Japanese ring
- Koszul complex of a module
- Koszul complex of a sequence of elements
- Krull's height theorem
- Krull's principal ideal theorem
- Krull-Azikuzi theorem
- Krull dimension
- Krull intersection theorem for Jacobson radical
- Krull intersection theorem for Noetherian domains
- Krull intersection theorem for modules
- Lagrange interpolation formula
- Laurent polynomial ring
- Leading coefficient map
- Length of irreducible factorization is strictly multiplicatively monotone on unique factorization domain
- Linearly closed subring
- Local Cohen-Macaulay ring
- Local Noetherian domain
- Local Noetherian domain implies equidimensional
- Local domain
- Local ring
- Localization at a prime ideal
- Localization at a submonoid
- Localization respects associated primes for Noetherian rings
- Locally a complete intersection
- Locally ringed space
- Main Page
- Map to localization is injective on spectra
- Max-spectrum of a commutative unital ring
- Max-spectrum of multivariate polynomial ring over an algebraically closed field
- Max-spectrum of polynomial ring over a field
- Maximal ideal
- Maximal implies prime
- Minimal prime ideal
- Minimal resolution
- Minimum over principal ideal of Euclidean norm is a smaller multiplicatively monotone Euclidean norm
- Module over a commutative unital ring
- Monomial ideal
- Morita-equivalent modules
- Morphism of varieties
- Multi-stage Euclidean domain
- Multi-stage Euclidean implies Bezout
- Multiplicative Dedekind-Hasse norm
- Multiplicative Euclidean norm
- Multiplicative norm
- Multiplicatively closed subset
- Multiplicatively monotone Euclidean norm
- Multiplicatively monotone Euclidean norm admits unique Euclidean division for exact divisor
- Multiplicatively monotone norm
- Multiplicatively monotone norm is constant on associate classes
- Multivariate polynomial ring
- Multivariate polynomial ring over a field
- Nagata ring
- Nakayama's lemma
- Natural map from topological space to max-spectrum of ring of continuous real-valued functions is an injection iff the space is Urysohn
- Nilpotent element
- Nilradical
- Nilradical equals intersection of all prime ideals
- Nilradical is an ideal
- Nilradical is smallest radical ideal
- Nilradical of subring lemma
- Noether normalization theorem
- Noetherian
- Noetherian UFD
- Noetherian and Bezout iff principal ideal
- Noetherian domain
- Noetherian domain implies every prime ideal is generated by finitely many irreducible elements
- Noetherian implies every element in minimal prime is zero divisor
- Noetherian implies intersection of minimal primes is irredundant
- Noetherian is polynomial-closed
- Noetherian local ring
- Noetherian local ring of positive dimension has element in maximal ideal outside minimal primes
- Noetherian module
- Noetherian normal domain
- Noetherian not implies zero divisor in minimal prime
- Noetherian ring
- Noetherian ring has finitely many minimal primes and every prime contains a minimal prime
- Noetherianness is quotient-closed
- Nonnegative norm
- Nonzerodivisor on a module
- Norm-Euclidean number field
- Norm-Euclidean ring of integers
- Norm on a commutative unital ring
- Normal
- Normal domain
- Normal ring
- Normality is polynomial-closed
- Normalization of a reduced ring
- Number field
- Number field with positive algebraic norm
- Numerical semigroup
- One-dimensional Noetherian domain implies Cohen-Macaulay
- One-dimensional domain
- One-dimensional ring
- PID implies UFD
- PID implies one-dimensional
- PID not implies Euclidean
- Perfect field
- Perfect ideal
- Permutation of regular sequence is not necessarily regular
- Polynomial-closed property
- Polynomial ring
- Polynomial ring over a field
- Polynomial ring over a field is uniquely Euclidean with norm equal to degree
- Polynomial ring over integrally closed subring is integrally closed in polynomial ring
- Power of an ideal
- Primary decomposition of an ideal
- Primary decomposition theorem for ideals
- Primary ideal
- Primary implies prime radical
- Primary ring
- Prime avoidance lemma
- Prime element
- Prime factorization of element is unique irreducible factorization upto ordering and associates
- Prime ideal
- Prime ideal need not contain any prime element
- Primeness is contraction-closed
- Principal ideal
- Principal ideal domain
- Principal ideal domain admits multiplicative Dedekind-Hasse norm
- Principal ideal is isomorphic to integral domain as a module
- Principal ideal ring
- Principal ideal ring iff every prime ideal is principal
- Principal ideal ring implies one-dimensional
- Principal prime ideal
- Product of ideals
- Projective algebraic variety
- Projective dimension of a module
- Projective module
- Projective resolution
- Proper ideal
- Proper integrally closed subring has infinite index
- Property of commutative unital rings
- Purely transcendental field extension
- Quadratic number field
- Quasi-Frobenius ring
- Quasilocal ring
- Quotient-closed property of commutative unital rings
- Rabinowitch's trick
- Radical ideal
- Radical of an ideal
- Radically closed subring
- Rational map of varieties
- Real quadratic number field
- Reduced Noetherian implies zero divisor in minimal prime
- Reduced Noetherian one-dimensional implies Cohen-Macaulay
- Reduced ring
- Regular function
- Regular local ring