Commutative algebra

theorem for solving simultaneous congruences

additive subgroup of a ring closed under multiplcation by arbitrary ring element

branch of algebra that studies commutative rings

algebraic structure

commutative ring with no zero divisors other than zero

ring of polynomials (with one or several variables) with coefficients in a given ring

Commutative ring with an Euclidean division

Algebraic structure

smallest field in which an integral domain can be embedded

ideal generated by one element

ring with unique factorization for ideals (mathematics)

algebra over a field

theorem

Algebraic definition of the dimension of a ring

mathematical formula

proper ideal q such that, whenever xy ∈ q, then either x ∈ q, or some power of y is in q

set of a ring's prime ideals

mathematical element

particular generating subset of an ideal in a polynomial ring

lemma

in algebra, expression of an ideal as the intersection of ideals of a specific type

theorem in commutative algebra

condition in commutative algebra

commutative ring, named after Irvin Cohen and Francis Sowerby Macaulay (1862-1937)

commutative Noetherian ring such that, at every prime ideal, the localization has finite injective dimension as a module over itself

set of the nilpotent elements of a ring

invariant of rings and modules in algebra

mathematical theorem of dimensional theory

polynomial with finitely many terms of the form axⁿ where n ∈ ℤ

in algebra, any of several related functors on rings and modules that result in complete topological rings and modules

principal ideal domain that is a local ring and not a field

integral domain in which the sum of two principal ideals is again a principal ideal

theorem

Concept in algebra

mathematical concept

theorem of analytic functions of several complex variables

module without any strictly ascending sequence of submodules

polynomial that only takes on integer values when evaluated at an integer

lemma stating that, given an ideal I in a Noetherian commutative ring R and a submodule N of a a finitely generated R-module M, there exists a positive integer k such that, for every n≥k, IⁿM ∩ N = Iⁿ⁻ᵏ(IᵏM ∩ N)

integral domain that is a certain kind of intersection of discrete valuation rings

ring in which every ideal is principal

commutative ring admitting a good dimension function, i.e. that relative dimension between two prime ideals is well defined, in the sense that any maximal strictly increasing chain of prime ideals between the two has the same finite length

power series with rational exponents

a valuation on a field taking values in the integers

Prime ideal involved in a primary decomposition

polynomial function with rational coefficients whose values agree, for sufficiently large argument, with the dimensions of graded components of a graded algebra

semihereditary integral domain

local ring in which Hensel’s lemma holds

a Noetherian universally catenary ring that is J-2 and Grothendieck

concept in mathematics

algebraically defined notion of differential form on an algebra over a ring or on a scheme

ring in algebra

mathematical domain whose elements share lowest common multiples

theorem in commutative algebra that every finitely generated projective module over a polynomial ring is free

differential associative algebra with integer grading in which the differential has grading +1 (cohomological convention) or −1 (homological convention)

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in commutative algebra, a sequence of elements of a commutative ring that is “as independent as possible” in a precise sense; the algebraic analogue of the geometric notion of a complete intersection

algorithm