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Endomorphism algebra of an abelian group
In mathematics, the endomorphisms of an abelian group X form a ring. This ring is called the endomorphism ring of X, denoted by End(X); the set of all
Endomorphism_ring
Self-self morphism
abstract algebra, an endomorphism is a homomorphism from a mathematical object to itself. More generally in category theory, an endomorphism is a morphism from
Endomorphism
Structure-preserving function between two rings
with ring homomorphisms as morphisms (see Category of rings). In particular, one obtains the notions of ring endomorphism, ring isomorphism, and ring automorphism
Ring_homomorphism
Theory of a class of elliptic curves
multiplication (CM) is the theory of elliptic curves E that have an endomorphism ring larger than the integers. Put another way, it contains the theory
Complex_multiplication
Algebraic structure with addition and multiplication
the endomorphisms of G form a ring, the endomorphism ring End(G) of G. The operations in this ring are addition and composition of endomorphisms. More
Ring_(mathematics)
In mathematics, element that equals its square
R (assumed unital), the endomorphism ring EndR(R) = R, where each endomorphism arises as left multiplication by a fixed ring element. With this modification
Idempotent_(ring_theory)
Homomorphisms between simple modules over the same ring are isomorphisms or zero
group ring of a finite group. However, even over the ring of integers, the module of rational numbers has an endomorphism ring that is a division ring, specifically
Schur's_lemma
Mathematical concept
case the endomorphism algebra has rank 4, while the endomorphism ring of every ordinary elliptic curve has rank 1 or 2. The endomorphism ring of a supersingular
Supersingular_elliptic_curve
Branch of algebra
mathematics. More generally, endomorphism rings of abelian groups are rarely commutative, the simplest example being the endomorphism ring of the Klein four-group
Ring_theory
(Mathematical) ring with a unique maximal ideal
Non-commutative local rings arise naturally as endomorphism rings in the study of direct sum decompositions of modules over some other rings. Specifically, if
Local_ring
Post-quantum digital signature scheme
elliptic curve is known as its endomorphism ring, written as End ( E ) {\displaystyle {\textrm {End}}(E)} . The endomorphism problem can be formulated as
SQIsign
Map raising elements to the pth power, in characteristic p
field theory, the Frobenius endomorphism (after Ferdinand Georg Frobenius) is a special endomorphism of commutative rings with prime characteristic p
Frobenius_endomorphism
Abstract algebra concept
states that if a module has an decomposition into modules with local endomorphism rings, then all decompositions into indecomposable modules are equivalent
Decomposition_of_a_module
algebra. Suppose M is a module over some ring. If M is indecomposable and has finite length, then every endomorphism of M is either an automorphism or nilpotent
Fitting_lemma
commutative domain is called an integral domain. endomorphism An endomorphism ring is a ring formed by the endomorphisms of an object with additive structure; the
Glossary_of_ring_theory
Unique ring consisting of one element
zero ring. The direct product of an empty collection of rings is the zero ring. The endomorphism ring of the trivial group is the zero ring. The ring of
Zero_ring
Direct sum of irreducible modules
homomorphism is a semiprimitive ring, and every semiprimitive ring is isomorphic to such an image. The endomorphism ring of a semisimple module is not only
Semisimple_module
Algebraic structure
(Lam 2001, §1,ex1.9). The skew-polynomial ring is defined similarly for a ring R and a ring endomorphism f of R, by extending the multiplication from
Polynomial_ring
Mathematical object in abstract algebra
multiplication by x behaves normally except that x·1 = 0. The endomorphism ring is simply the ring of formal power series. If G is a finite group and k a field
Injective_module
In mathematics, invariant of square matrices
allows defining the determinant of an endomorphism as the determinant of the matrix that represents the endomorphism on any basis. Indeed, this determinant
Determinant
Type of module over a ring
homomorphism or an isomorphism. Consequently, the endomorphism ring of any simple module is a division ring. This result is known as Schur's lemma. The converse
Simple_module
Algebraic structure generalizing Boolean rings
and 1. The endomorphism ring of a continuous module is a clean ring. Every clean ring is an exchange ring. A matrix ring over a clean ring is itself clean
Clean_ring
direct summand is itself a direct summand. The endomorphism ring of a continuous module is a clean ring. Camillo, V.P.; Khurana, D.; Lam, T.Y.; Nicholson
Continuous_module
Linear map over a ring
multiplication given by function composition, called the endomorphism ring of M. The group of units of this ring is the automorphism group of M. Schur's lemma says
Module_homomorphism
Mathematical ring with well-behaved ideals
of H. The endomorphism ring of an indecomposable injective module is local and thus Azumaya's theorem says that, over a left Noetherian ring, each indecomposable
Noetherian_ring
Algebraic structure also called skew field
division ring. In general, if R is a ring and S is a simple module over R, then, by Schur's lemma, the endomorphism ring of S is a division ring; every
Division_ring
Algebraic structure in linear algebra
multiplication) say that this operation defines a ring homomorphism from the field F into the endomorphism ring of this group. Specifically, the distributivity
Vector_space
Algebraic structure
mathematics, a commutative ring is a ring in which the multiplication operation is commutative. The study of commutative rings is called commutative algebra
Commutative_ring
idempotent endomorphism of M, then M is the direct sum of ker(f) and im(f).) A module of finite length is indecomposable if and only if its endomorphism ring is
Indecomposable_module
finds applications especially in the study of endomorphism rings where we have A = B. Similarly, if R is a ring and M is a right R-module, then the finite
Finite_topology
Japanese mathematician (born 1951)
won the Fields Medal in 1990. Mori completed his Ph.D. titled "The Endomorphism Rings of Some Abelian Varieties" under Masayoshi Nagata at Kyoto University
Shigefumi_Mori
Array of numbers
n-by-n matrices over R is a ring called matrix ring, isomorphic to the endomorphism ring of the left R-module Rn. If the ring R is commutative, that is
Matrix_(mathematics)
Rings admitting weak inverses
the endomorphism ring EndS(M) is von Neumann regular. In particular, every semisimple ring is von Neumann regular. Indeed, the semisimple rings are precisely
Von_Neumann_regular_ring
Submodule of a mathematical ring
In mathematics, and more specifically in ring theory, an ideal of a ring is a special subset of its elements. Ideals generalize certain subsets of the
Ideal_(ring_theory)
Algebraic structure used in analysis
associativity of the multiplication on A {\displaystyle A} .) The endomorphism ring of an F {\displaystyle F} -vector space V {\displaystyle V} with the
Lie_algebra
Mathematical term in group theory
every Artinian ring is Noetherian). The endomorphism ring of Z ( p ∞ ) {\displaystyle \mathbb {Z} (p^{\infty })} is isomorphic to the ring of p-adic integers
Prüfer_group
Reduction of a ring by one of its ideals
In ring theory, a branch of abstract algebra, a quotient ring, also known as factor ring, difference ring or residue class ring, is a construction quite
Quotient_ring
Mathematical group that can be generated as the set of powers of a single element
the endomorphism ring of the additive group of Z is isomorphic to the ring Z. Its automorphism group is isomorphic to the group of units of the ring Z,
Cyclic_group
Mathematical function between groups that preserves multiplication structure
elements except identity. Endomorphism A group homomorphism, h: G → G; the domain and codomain are the same. Also called an endomorphism of G. Automorphism A
Group_homomorphism
Algebraic structure with "nice" duality properties
unital associative algebra A has a natural homomorphism to its own endomorphism ring End(A). A bilinear form can be defined on A in the sense of the previous
Frobenius_algebra
ring R. We call C a Krull–Schmidt category provided that every object decomposes into a finite direct sum of objects having local endomorphism rings.
Krull–Schmidt_category
Mathematical element
normal, the endomorphism ring S = Hom A ( I , I ) {\displaystyle S=\operatorname {Hom} _{A}(I,I)} provides a strictly larger integral ring extension A
Integral_element
Ring that is also a vector space or a module
Z-modules are equivalent. Any ring of characteristic n is a (Z/nZ)-algebra in the same way. Given an R-module M, the endomorphism ring of M, denoted EndR(M) is
Associative_algebra
Algebraic ring that need not have additive negative elements
a semiring is an algebraic structure. Semirings are a generalization of rings, dropping the requirement that each element must have an additive inverse
Semiring
Mathematical structure in abstract algebra
an involution is important to the Kazhdan–Lusztig polynomial. The endomorphism ring of an elliptic curve becomes a *-algebra over the integers, where
*-algebra
Isomorphism of an object to itself
Antiautomorphism Automorphism (in Sudoku puzzles) Characteristic subgroup Endomorphism ring Frobenius automorphism Morphism Order automorphism (in order theory)
Automorphism
Algebraic construction
In mathematics, the ring of integers of an algebraic number field K {\displaystyle K} (also sometimes called the number ring corresponding to number field
Ring_of_integers
Mathematical ideal related to a modular curve
In mathematics, the Eisenstein ideal is an ideal in the endomorphism ring of the Jacobian variety of a modular curve, consisting roughly of elements of
Eisenstein_ideal
Project by NIST to standardize post-quantum cryptography
Supersingular elliptic curve isogeny SQIsign Fiat–Shamir heuristic Endomorphism Ring Problem Symmetric-based FAEST "in the head", Fiat–Shamir heuristic
NIST Post-Quantum Cryptography Standardization
NIST_Post-Quantum_Cryptography_Standardization
Prime number with a certain relationship to an elliptic curve
{\displaystyle E} modulo p {\displaystyle p} has the maximum possible endomorphism ring—an order in a quaternion algebra—rather than an order in an imaginary
Supersingular prime (algebraic number theory)
Supersingular_prime_(algebraic_number_theory)
In mathematics, the Auslander algebra of an algebra A is the endomorphism ring of the sum of the indecomposable modules of A. It was introduced by Auslander (1974)
Auslander_algebra
Classification of semi-simple rings and algebras
(I_{i}){\big )}} where the endomorphism ring E n d ( I i ) {\displaystyle \mathrm {End} (I_{i})} of I i {\displaystyle I_{i}} is a division ring by Schur's lemma
Wedderburn–Artin_theorem
Mathematical concept
illustrates that Dedekind-finite rings need not be closed under homomorpic images. Another non-example is the endomorphism ring End ( V ) {\displaystyle \operatorname
Dedekind-finite_ring
Algebraic ring classification
indeed a semisimple ring. The classical ring of quotients for any commutative Noetherian ring is a semilocal ring. The endomorphism ring of an Artinian module
Semi-local_ring
Commutative ring with no zero divisors other than zero
In mathematics, an integral domain is a nonzero commutative ring in which the product of any two nonzero elements is nonzero. In an integral domain, every
Integral_domain
Branch of mathematics
inverses. A linear endomorphism is a linear map that maps a vector space V to itself. If V has a basis of n elements, such an endomorphism is represented
Linear_algebra
Algebraic structure
Equivalently, a noncommutative ring is a ring that is not a commutative ring. Noncommutative algebra is the part of ring theory devoted to study of properties
Noncommutative_ring
Vector space equipped with a bilinear product
associativity is not assumed (but not excluded, either). Given an integer n, the ring of real square matrices of order n is an example of an associative algebra
Algebra_over_a_field
Generalization of vector spaces from fields to rings
necessarily a group endomorphism of the abelian group (M, +). The set of all group endomorphisms of M is denoted EndZ(M) and forms a ring under addition and
Module_(mathematics)
Algebra over a field with only invertible elements and zero
algebra over the field F and S is a simple module over A, then the endomorphism ring of S is a division algebra over F; every associative division algebra
Division_algebra
Studies linear representations of finite groups over fields of positive characteristic
sufficiently large: each block is a full matrix algebra over F, the endomorphism ring of the vector space underlying the associated simple module. To obtain
Modular_representation_theory
Mathematical ring whose elements are matrices
composition of endomorphisms. The ring Mn(D) over a division ring D is an Artinian simple ring, a special type of semisimple ring. The rings C F M I ( D
Matrix_ring
known as ring theory, a left primitive ring is a ring which has a faithful simple left module. Well known examples include endomorphism rings of vector
Primitive_ring
Topics referred to by the same term
A precise meaning depends on context: A semisimple element in the endomorphism ring of a vector space is a semisimple operator. In a semisimple Lie algebra
Semisimple_element
Algebraic structure in mathematics
mathematics, a near-ring (also near ring or nearring) is an algebraic structure similar to a ring but satisfying fewer axioms. Near-rings arise naturally
Near-ring
tower of iterated endomorphism rings above the subring. A more recent definition of depth of any unital subring in any associative ring is proposed (see
Depth of noncommutative subrings
Depth_of_noncommutative_subrings
certain endomorphism ring, and the ring structure is taken across this isomorphism to imbue Ẽ(R) with a ring structure, that of the maximal right ring of quotients
Dense_submodule
since the endomorphism ring of a countably infinite dimensional vector space is semiprimitive, but not a subdirect product of simple rings, (Lam 1995
Semiprimitive_ring
Infinite sum that is considered independently from any notion of convergence
} called coefficients, are numbers or, more generally, elements of some ring, and the x n {\displaystyle x^{n}} are formal powers of the symbol x {\displaystyle
Formal_power_series
elementary elementary divisor endomorphism 1. An endomorphism is a module homomorphism from a module to itself. 2. The endomorphism ring is the set of all module
Glossary_of_module_theory
Ring in abstract algebra
k\cdot y^{2}} is an Artinian ring with maximal ideal ( x , y ) {\displaystyle (x,y)} . Let x {\displaystyle x} be an endomorphism between a finite-dimensional
Artinian_ring
Algebraic structure in mathematics
The maximal ring of quotients Q(R) (in the sense of Utumi and Lambek) of a Boolean ring R is a Boolean ring, since every partial endomorphism is idempotent
Boolean_ring
Mathematical category whose hom sets form Abelian groups
composition. This ring is the endomorphism ring of A {\displaystyle A} . Conversely, every ring (with identity) is the endomorphism ring of some object in
Preadditive_category
Mathematical theorem in representation theory
{\displaystyle B} is the centralizer of A {\displaystyle A} in the endomorphism ring End C ( U ) {\displaystyle \operatorname {End} _{\mathbb {C} }(U)}
Schur–Weyl_duality
Non-tensorial representation of the spin group
so extends to a homomorphism of the full Clifford algebra into the endomorphism ring End(Δ) by the universal property of Clifford algebras. The details
Spinor
Weil cohomology theory for schemes X over a base field k
if X {\displaystyle X} is a supersingular elliptic curve, then its endomorphism ring is a maximal order in a quaternion algebra B {\displaystyle B} over
Crystalline_cohomology
original short proof uses generic flatness. Quillen, D. (1969). "On the endomorphism ring of a simple module over an enveloping algebra". Proceedings of the
Quillen's_lemma
Differential mapping
is a ring with a p-derivation, then the map σ ( x ) := x p + p δ ( x ) {\displaystyle \sigma (x):=x^{p}+p\delta (x)} defines a ring endomorphism which
P-derivation
Subset of a ring that forms a ring itself
In mathematics, a subring of a ring R is a subset of R that is itself a ring when binary operations of addition and multiplication on R are restricted
Subring
On generating functions from counting points on algebraic varieties over finite fields
supersingular elliptic curve over a finite field of characteristic p. The endomorphism ring of this is an order in a quaternion algebra over the rationals, and
Weil_conjectures
Branch of algebra that studies commutative rings
commutative algebra. Prominent examples of commutative rings include polynomial rings; rings of algebraic integers, including the ordinary integers Z
Commutative_algebra
Mathematical expression for linear operators
_{\mathbb {Q} }(k)} the endomorphism ring of k over rational numbers and V a finite-dimensional vector space over k. Given an endomorphism x : V → V {\displaystyle
Jordan–Chevalley decomposition
Jordan–Chevalley_decomposition
Abelian categories, while abstractly defined, are in fact concrete categories of modules
{\displaystyle I} . The endomorphism ring R := Hom L ( I , I ) {\displaystyle R:=\operatorname {Hom} _{\mathcal {L}}(I,I)} is the ring we need for the category
Mitchell's_embedding_theorem
Mathematical property
finitely many simple objects. It follows from Schur's lemma that the endomorphism ring End C ( X ) = Hom C ( X , X ) {\displaystyle \operatorname {End}
Semi-simplicity
Projective variety that is also an algebraic group
Schottky problem. A polarisation induces a Rosati involution on the endomorphism ring E n d ( A ) ⊗ Q {\displaystyle \mathrm {End} (A)\otimes \mathbb {Q}
Abelian_variety
Kind of complex manifold
homomorphisms. These are useful to determining some information about the endomorphism ring End ( X ) ⊗ Q {\displaystyle {\text{End}}(X)\otimes \mathbb {Q} }
Complex_torus
have CM-type if it has a large enough commutative subring in its endomorphism ring End(A). The terminology here is from complex multiplication theory
Complex multiplication of abelian varieties
Complex_multiplication_of_abelian_varieties
Branch of mathematics that studies algebraic structures
rings Quotient ring Matrix ring Endomorphism ring Polynomial ring Formal power series Monoid ring, Group ring Localization of a ring Tensor algebra Symmetric
List of abstract algebra topics
List_of_abstract_algebra_topics
Construction within abstract algebra
quotient ring or total ring of fractions is a construction that generalizes the notion of the field of fractions of an integral domain to commutative rings R
Total_ring_of_fractions
Type of Abelian category (in category theory in mathematics)
(R)} of right modules over some unital ring R {\displaystyle R} (which can be taken to be the endomorphism ring of a generator of A {\displaystyle {\mathcal
Grothendieck_category
Mathematical concept
discovered that elliptic curves over finite fields can have even larger endomorphism rings of rank 4, and these were termed "supersingular elliptic curves".
Supersingular_variety
right R-module. The endomorphism ring S = E n d ( E ( R R ) ) {\displaystyle S=\mathrm {End} (E(R_{R}))\,} is a semiprimitive ring (that is, J ( S ) =
Singular_submodule
Bézout module. It is known that the endomorphism ring EndR M is a semilocal ring which is very close to a local ring in the sense that EndR M has at most
Serial_module
Algebraic structure
mathematics, the ring of polynomial functions on a vector space V over a field k gives a coordinate-free analog of a polynomial ring. It is denoted by
Ring_of_polynomial_functions
Conjecture in number theory
surfaces defined over Q {\displaystyle \mathbb {Q} } with geometric endomorphism ring equal to a maximal order in a non-split quaternion algebra, Jef Laga
Torsion_conjecture
itself is the identity endomorphism of the space of 2-forms, Ω2(M), under the identification (using the metric) of the endomorphism ring End(Ω2(M)) with the
Kulkarni–Nomizu_product
Category whose objects are rings and whose morphisms are ring homomorphisms
mathematics, the category of rings, denoted by Ring, is the category whose objects are rings (with identity) and whose morphisms are ring homomorphisms (that preserve
Category_of_rings
Mathematical function, in linear algebra
same, a linear map is also called a linear transformation or linear endomorphism. A linear map is a homomorphism of vector spaces. Thus, a linear map
Linear_map
Set of finitely supported functions from a group to a ring
ring is a free module and at the same time a ring, constructed in a natural way from any given ring and any given group. As a free module, its ring of
Group_ring
Type of category in category theory
Recall that the morphisms from a single object A to itself form the endomorphism ring End A. If we denote the n-fold product of A with itself by An, then
Additive_category
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ENDOMORPHISM RING
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ENDOMORPHISM RING
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