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Branch of algebra that studies commutative rings
Commutative algebra, first known as ideal theory, is the branch of algebra that studies commutative rings, their ideals, and modules over such rings.
Commutative_algebra
Field of mathematics using techniques from combinatorics and commutative algebra
Combinatorial commutative algebra is a relatively new, rapidly developing mathematical discipline. As the name implies, it lies at the intersection of
Combinatorial commutative algebra
Combinatorial_commutative_algebra
Mathematical ring
Stanley–Reisner ring construction is a basic tool within algebraic combinatorics and combinatorial commutative algebra. Its properties were investigated by Richard
Stanley–Reisner_ring
Algebraic structure with addition and multiplication
A commutative ring is a ring with a commutative multiplication. This property has profound implications on ring properties. Commutative algebra, the
Ring_(mathematics)
Area of combinatorics
geometries. Algebraic graph theory Combinatorial commutative algebra Polyhedral combinatorics Algebraic Combinatorics (journal) Journal of Algebraic Combinatorics
Algebraic_combinatorics
Branch of discrete mathematics
breadth of the problems it tackles. Combinatorial problems arise in many areas of pure mathematics, notably in algebra, probability theory, topology, and
Combinatorics
Cohomology theory Combinatorial analysis Combinatorial commutative algebra a discipline viewed as the intersection between commutative algebra and combinatorics
Glossary of areas of mathematics
Glossary_of_areas_of_mathematics
Convex hull of points on moment curve
homological methods. Combinatorial commutative algebra Miller, Ezra; Sturmfels, Bernd (2005). Combinatorial commutative algebra. Graduate Texts in Mathematics
Cyclic_polytope
Commutative algebra studies commutative rings, their ideals, and modules over such rings
of algebraic geometry, and many results and concepts of commutative algebra are strongly related with geometrical concepts. Combinatorial commutative algebra
List of commutative algebra topics
List_of_commutative_algebra_topics
Branch of mathematics
geometry, that studies the geometric properties of formal duals of non-commutative algebraic objects such as rings as well as geometric objects derived from
Noncommutative algebraic geometry
Noncommutative_algebraic_geometry
Algebraic ring that need not have additive negative elements
isomorphic to a sub-semiring of a Boolean algebra. The commutative semiring formed by the two-element Boolean algebra and defined by 1 + 1 = 1 {\displaystyle
Semiring
Branch of mathematics
can be traced to investigations in combinatorial topology (a precursor to algebraic topology) and abstract algebra (theory of modules and syzygies) at
Homological_algebra
American mathematician and violinist
the AWM–Microsoft Research Prize in Algebra and Number Theory. Her research involves combinatorial commutative algebra, graph theory, and tropical geometry
Melody_Chan
Graph of zero divisors of a commutative ring
specifically in combinatorial commutative algebra, a zero-divisor graph is an undirected graph representing the zero divisors of a commutative ring. It has
Zero-divisor_graph
In mathematics, invariant of square matrices
classes of matrices with non-commutative elements, one can prove linear algebra theorems that are very similar to their commutative analogs. Examples include
Determinant
Branch of mathematics
Algebraic geometry is a branch of mathematics which uses abstract algebraic techniques, mainly from commutative algebra, to solve geometrical problems
Algebraic_geometry
Ideal generated by differences of monomials
projective toric variety. Miller, Ezra; Sturmfels, Bernd (2005), Combinatorial Commutative Algebra, Graduate Texts in Mathematics, vol. 227, New York: Springer-Verlag
Toric_ideal
Associative algebra used in combinatorics
mathematics, an incidence algebra is an associative algebra, defined for every locally finite partially ordered set and commutative ring with unity. Subalgebras
Incidence_algebra
Branch of mathematics
ideas through noncommutative algebras. In ordinary geometry, a space can often be studied by means of a commutative algebra of functions on it; noncommutative
Noncommutative_geometry
In mathematics, a Bose–Mesner algebra is a special set of matrices which arise from a combinatorial structure known as an association scheme, together
Bose–Mesner_algebra
Mathematical theory
Press, 2001 [1994] Miller, Ezra; Sturmfels, Bernd (2005). Combinatorial Commutative Algebra. Graduate Texts in Mathematics. Vol. 227. New York, NY: Springer-Verlag
Alexander_duality
Four-dimensional number system
normed division algebra over the real numbers, and therefore a ring, also a division ring and a domain. Because of their non-commutative multiplication
Quaternion
Algebra used in 2D conformal field theories and string theory
Other formulations of the vertex algebra axioms include Borcherds's later work on singular commutative rings, algebras over certain operads on curves introduced
Vertex_operator_algebra
Type of polytope in mathematics
In mathematics, specifically in combinatorial commutative algebra, a convex lattice polytope P is called normal if it has the following property: given
Normal_polytope
Numbering of combinations of items
and Green's hyperplane restriction theorem", Commutative Algebra: Geometric, Homological, Combinatorial and Computational Aspects, CRC Press, ISBN 978-1-420-02832-4
Combinatorial_number_system
Algebra in statistical mechanics
von Neumann algebras. Let R {\displaystyle R} be a commutative ring and fix δ ∈ R {\displaystyle \delta \in R} . The Temperley–Lieb algebra T L n ( δ )
Temperley–Lieb_algebra
Symmetric arrangement of finite sets
association schemes, yielding the field of algebraic statistics. The classical core of the subject of combinatorial designs is built around balanced incomplete
Combinatorial_design
Study of systems of inequalitites
The relation of real algebra to real algebraic geometry is similar to the relation of commutative algebra to complex algebraic geometry. Related fields
Real_algebraic_geometry
Israeli mathematician (1921–1994)
general ring theory, structure theory of PI-rings, combinatorial PI-theory, and division algebras. After the death of his advisor Jacob Levitzki in 1956
Shimshon_Amitsur
secondary-school mathematics textbooks Melody Chan, American expert in combinatorial commutative algebra, graph theory, and tropical geometry Sun-Yung Alice Chang
List_of_women_in_mathematics
1016/0001-8708(78)90037-3. Miller, Ezra; Sturmfels, Bernd (2005). Combinatorial Commutative Algebra. Graduate Texts in Mathematics. Vol. 227. Springer. ISBN 978-0-387-23707-7
Determinantal_variety
C*-algebra
defined and described combinatorially by Ola Bratteli. Later, George A. Elliott gave a complete classification of AF algebras using the K0 functor whose
Approximately finite-dimensional C*-algebra
Approximately_finite-dimensional_C*-algebra
In commutative algebra, a Rees decomposition is a way of writing a ring in terms of polynomial subrings. They were introduced by David Rees (1956). Suppose
Rees_decomposition
Theory in statistics
both algebra and combinatorics. In algebraic combinatorics, association schemes provide a unified approach to many topics, for example combinatorial designs
Association_scheme
American mathematician (born 1944)
Combinatorics (1986–1999). He is also the author of Combinatorics and Commutative Algebra (1983) and well over 200 research articles in mathematics. He has
Richard_P._Stanley
of Combinatorial Theory, Series B. 10 (3): 187–200. doi:10.1016/0095-8956(71)90042-6. Stanley, Richard (1996). Combinatorics and commutative algebra. Progress
Simplicial_sphere
Generalization of associativity properties
trivially. The algebras over this operad are the commutative semigroups; the k-linear algebras are the commutative associative k-algebras. Similarly, there
Operad
In commutative algebra, a Stanley decomposition is a way of writing a ring in terms of polynomial subrings. They were introduced by Richard Stanley (1982)
Stanley_decomposition
Branch of mathematical statistics
Algebraic statistics is a branch of mathematical statistics that focuses on the use of algebraic, geometric, and combinatorial methods in statistics. While
Algebraic_statistics
Array of numbers
Rn. If the ring R is commutative, that is, its multiplication is commutative, then the ring M(n, R) is also an associative algebra over R. The determinant
Matrix_(mathematics)
Number used for counting
produce b {\displaystyle b} . The algebraic structure ( N , + ) {\displaystyle (\mathbb {N} ,+)} is a commutative monoid with identity element 0 {\displaystyle
Natural_number
Number of subsets of a given size
(if k ≤ n) in the binomial formula (valid for any elements x, y of a commutative ring), which explains the name "binomial coefficient". Another occurrence
Binomial_coefficient
the connected components of M-curves? Homological conjectures in commutative algebra Jacobson's conjecture: the intersection of all powers of the Jacobson
List of unsolved problems in mathematics
List_of_unsolved_problems_in_mathematics
Series of mathematics textbooks
in the Unit Ball, Kehe Zhu, (2005, ISBN 978-0-387-22036-9) Combinatorial Commutative Algebra, Ezra Miller, Bernd Sturmfels, (2005, ISBN 978-0-387-22356-8)
Graduate_Texts_in_Mathematics
theorems (commutative algebra) Hilbert's basis theorem (commutative algebra,invariant theory) Hilbert's syzygy theorem (commutative algebra) Integral
List_of_theorems
Algebraic topology uses abstract algebra to study topological spaces
K-theory Hodge conjecture Weil conjectures Directed algebraic topology Example: DE-9IM Chain complex Commutative diagram Exact sequence Five lemma Short five
List of algebraic topology topics
List_of_algebraic_topology_topics
Square matrices satisfy their characteristic equation
theorem is the source of the celebrated Nakayama lemma in commutative algebra and algebraic geometry. The Cayley-Hamilton theorem also holds for matrices
Cayley–Hamilton_theorem
American mathematician
http://library.msri.org/books/Book38/ Simplicial complex Combinatorial commutative algebra Quasisymmetric function Louis Billera at the Mathematics Genealogy
Louis_Billera
Embedding of a Grassmannian into projective space
MR 1288523, Zbl 0836.14001 Miller, Ezra; Sturmfels, Bernd (2005). Combinatorial commutative algebra. Graduate Texts in Mathematics. Vol. 227. New York, NY: Springer-Verlag
Plücker_embedding
Algebraic study of differential equations
integers in number theory Difference algebra Differential algebraic geometry Differential calculus over commutative algebras Differential Galois theory – Study
Differential_algebra
German mathematician (1882–1935)
commutative ring theory, and gives one of the first general definitions of a commutative ring. Before her paper, most results in commutative algebra were
Emmy_Noether
Concepts from linear algebra
In linear algebra, an eigenvector (/ˈaɪɡən-/ EYE-gən-) or characteristic vector is a (nonzero) vector that has its direction unchanged (or reversed) by
Eigenvalues_and_eigenvectors
"The Hopf algebras of symmetric functions and quasi-symmetric functions in non-commutative variables are free and co-free", Journal of Algebra and Its Applications
Quasisymmetric_function
diagrams can be described by a Hopf algebra. Combinatorial physics can be characterized by the use of algebraic concepts to interpret and solve physical
Combinatorics_and_physics
k} be a commutative ring and let λ {\displaystyle \lambda } be given. A linear operator R {\displaystyle R} on a k {\displaystyle k} -algebra A {\displaystyle
Rota–Baxter_algebra
(polynomials) Schwartz–Zippel lemma Artin–Rees lemma Hensel's lemma (commutative rings) Nakayama lemma Noether's normalization lemma Prime avoidance lemma
List_of_lemmas
In mathematics, a median algebra is a set with a ternary operation ⟨ x , y , z ⟩ {\displaystyle \langle x,y,z\rangle } satisfying a set of axioms which
Median_algebra
Relation between algebraic varieties and polynomial ideals
Introduction to Algebraic Geometry and Commutative Algebra. World Scientific. ISBN 978-9814307581. Reid, Miles (1995). Undergraduate commutative algebra. London
Hilbert's_Nullstellensatz
American mathematician
Morey is known for her work in commutative algebra, in particular, for work on normal rings and algebraic and combinatorial properties of edge ideals of
Susan_Morey
Theoretical object in mathematics
their abstract properties. This allows the development of commutative algebra and algebraic geometry on new foundations. One of the defining features
Field_with_one_element
Branch of mathematics
methods—differential geometry, algebraic geometry, computational geometry, algebraic topology, discrete geometry (also known as combinatorial geometry), etc.—or on
Geometry
Arithmetic operation
the commutative ring is said to be reduced. Reduced rings are important in algebraic geometry, since the coordinate ring of an affine algebraic set is
Exponentiation
Concept in mathematics
In abstract algebra, the free monoid on a set is the monoid whose elements are all the finite sequences (or strings) of zero or more elements from that
Free_monoid
Branch of mathematics that studies abstract algebraic structures
dual vector space. The Hopf algebras associated to groups have a commutative algebra structure, and so general Hopf algebras are known as quantum groups
Representation_theory
Iranian mathematician (born 1959)
born 1959) is an Iranian mathematician specializing in commutative algebra, homological algebra, and combinatorics. He is an Associate Professor of Practice
Siamak_Yassemi
Serbian-American statistician
Kentucky in Lexington, Kentucky, specializing in commutative algebra. Her dissertation Algebraic and Combinatorial Properties of Certain Toric Ideals in the
Sonja_Petrović_(statistician)
American mathematician
domain and analytic subvarieties. D'Angelo's work uses analysis and commutative algebra. A domain all of whose boundary points have finite D'Angelo type
John_D'Angelo
Largest integer that divides given integers
1016/0022-314X(87)90081-3. Lovett, Stephen (2015). "Divisibility in Commutative Rings". Abstract Algebra: Structures and Applications. Boca Raton: CRC Press. pp. 267–318
Greatest_common_divisor
Number line and triangular tiling's symmetry mathematical structure
+ n {\displaystyle 1+2+\cdots +n} . To translate between the combinatorial and algebraic definitions, for i = 1 , … , n − 1 {\displaystyle i=1,\ldots
Affine_symmetric_group
Application of group theory to cryptography
(help) Shpilrain, V.; Zapata, G. (2006). "Combinatorial group theory and public key cryptography". Appl. Algebra Eng. Commun. Comput. 17 (3–4): 291–302.
Group-based_cryptography
theory topics List of cohomology theories List of commutative algebra topics List of homological algebra topics List of group theory topics Glossary of group
Lists_of_mathematics_topics
Combinitorics of Polyhedra
descriptions of their facets are available. Abstract polytope Combinatorial commutative algebra Matroid polytope Order polytope Simplicial sphere Stable matching
Polyhedral_combinatorics
Magma obeying the Latin square property
x2y1)). Then, (F4, ∗) is a commutative Moufang loop that is not a group. More generally, the nonzero elements of any division algebra form a quasigroup with
Quasigroup
Theory in mathematics
In combinatorial mathematics, the theory of combinatorial species is an abstract, systematic method for deriving the generating functions of discrete structures
Combinatorial_species
Overview of and topical guide to category theory
Saunders Mac Lane in the mid-20th century in their foundational work on algebraic topology. Category theory can be used in most areas of mathematics. In
Outline_of_category_theory
Ideal generated by one-term polynomials
algebra Stanley–Reisner ring Toric ideal Dummit & Foote 2004 Miller & Sturmfels 2005 Miller, Ezra; Sturmfels, Bernd (2005), Combinatorial Commutative
Monomial_ideal
Mathematical theory on random variables
Free probability is a mathematical theory that studies non-commutative random variables. The "freeness" or free independence property is the analogue
Free_probability
Professor of mathematics
University of Warwick. She is a researcher in combinatorial and computational commutative algebra and algebraic geometry, with an emphasis on toric varieties
Diane_Maclagan
Field of knowledge
theory commutative algebra, which is the study of commutative rings, includes the study of polynomials, and is a foundational part of algebraic geometry
Mathematics
In abstract algebra, an Artinian ideal, named after Emil Artin, is encountered in ring theory, in particular, with polynomial rings. Given a polynomial
Artinian_ideal
0112.06 Wu, Yu-Fen (1998), "Groups in which commutativity is a transitive relation", Journal of Algebra, 207 (1): 165–181, doi:10.1006/jabr.1998.7468
CA-group
Linear operator acting on modular forms
Hecke operators. Algebras of Hecke operators are called "Hecke algebras", and are commutative rings. In the classical elliptic modular form theory, the Hecke
Hecke_operator
States that the algebra of n by n matrices satisfies a certain identity of degree 2n
In algebra, the Amitsur–Levitzki theorem states that the algebra of n × n matrices over a commutative ring satisfies a certain identity of degree 2n. It
Amitsur–Levitzki_theorem
Branch of algebraic geometry
abelian group. Modern foundations of algebraic geometry were developed based on contemporary commutative algebra, including valuation theory and the theory
Arithmetic_geometry
Integer matrices with +1 or −1 determinant; invertible over the integers. GL_n(Z)
submatrix of determinant −2. Abstract linear algebra considers matrices with entries from any commutative ring R {\displaystyle R} , not limited to the
Unimodular_matrix
Graphical representation of a morphism
and low-dimensional topology, a combinatorial definition is necessary to formalise string diagrams in computer algebra systems and use them to define computational
String_diagram
Mathematical object
be studied algebraically by forming its Stanley–Reisner ring; this sets up a powerful relation between combinatorics and commutative algebra. A collection
Abstract_simplicial_complex
Concept in topology
Hindman and Strauss) guarantees that these central sets are combinatorially rich. For a commutative semigroup S, the Central Sets Theorem states that if A
Stone–Čech_compactification
Concept in algebraic geometry
Chapter 10, Residue Methods in Combinatorial Analysis) Stanley, Richard (1996). Combinatorics and commutative algebra. Boston, MA: Birkhäuser Boston,
Local_cohomology
Mathematical set closed under positive linear combinations
"Rational cones are important objects in toric algebraic geometry, combinatorial commutative algebra, geometric combinatorics, integer programming."
Convex_cone
String that is strictly smaller in lexicographic order than all of its rotations
of characteristic 0 (or, more general, a commutative ℚ-algebra), and let R be the free noncommutative k-algebra k ⟨ xa | a ∈ A ⟩. The words over A can then
Lyndon_word
Russian mathematician
secant variety: on a theorem of G. Scorza", Geometric and combinatorial aspects of commutative algebra (Messina, 1999), Lecture Notes in Pure and Appl. Math
Fyodor_Zak
Awarded every year by the American Mathematical Society
Society. ISBN 9780821853368. Eisenbud, David (1995). Commutative Algebra with a View Toward Algebraic Geometry. Graduate Texts in Mathematics. Vol. 150.
Leroy_P._Steele_Prize
semigroups by seeking relationships between the combinatorial structure of the graph and the algebraic structure of the group or semigroup. Depending on
Commuting_graph
They then apply this to the group algebras of G and H over any commutative ring R. They define a minimum combinatorial depth d c ( H , G ) {\displaystyle
Depth of noncommutative subrings
Depth_of_noncommutative_subrings
Homological algebra is the study of homological functors
can be traced to investigations in combinatorial topology (a precursor to algebraic topology) and abstract algebra (theory of modules and syzygies) at
List of homological algebra topics
List_of_homological_algebra_topics
General concept and operation in mathematics
tells that the local theory of schemes is the same as commutative algebra, the study of commutative rings. Noncommutative geometry draws inspiration from
Duality_(mathematics)
Used to count, measure, and label
Holweck, Frédéric; Pracna, Petr (2015). "From Cayley-Dickson Algebras to Combinatorial Grassmannians". Mathematics. 3 (4). MDPI AG: 1192–1221. arXiv:1405
Number
About the numbers of faces of different dimensions in an abstract simplicial complex
1016/j.disc.2019.111801 Stanley, Richard (1996), Combinatorics and commutative algebra, Progress in Mathematics, vol. 41 (2nd ed.), Boston, MA: Birkhäuser
Kruskal–Katona_theorem
About polynomials in several variables
Shpilrain, Vladimir (1997). "Some combinatorial questions about polynomial mappings". Journal of Pure and Applied Algebra. 119: 47–52. doi:10.1016/S0022-4049(96)00043-6
Jacobian_conjecture
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COMBINATORIAL COMMUTATIVE-ALGEBRA
COMBINATORIAL COMMUTATIVE-ALGEBRA
COMBINATORIAL COMMUTATIVE-ALGEBRA
COMBINATORIAL COMMUTATIVE-ALGEBRA
COMBINATORIAL COMMUTATIVE-ALGEBRA
COMBINATORIAL COMMUTATIVE-ALGEBRA
COMBINATORIAL COMMUTATIVE-ALGEBRA
COMBINATORIAL COMMUTATIVE-ALGEBRA
COMBINATORIAL COMMUTATIVE-ALGEBRA
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