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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
Endomorphism algebra of an abelian group
under consideration. The endomorphism ring consequently encodes several internal properties of the object. As the endomorphism ring is often an algebra
Endomorphism_ring
Map raising elements to the pth power, in characteristic p
algebra and field theory, the Frobenius endomorphism (after Ferdinand Georg Frobenius) is a special endomorphism of commutative rings with prime characteristic
Frobenius_endomorphism
Mathematical function, in linear algebra
transformation f : V → V {\textstyle f:V\to V} is an endomorphism of V {\textstyle V} ; the set of all such endomorphisms End ( V ) {\textstyle \operatorname {End}
Linear_map
Structure-preserving function between two rings
of prime characteristic p, R → R, x → xp is a ring endomorphism called the Frobenius endomorphism. If R and S are rings, the zero function from R to S
Ring_homomorphism
Structure-preserving map between two algebraic structures of the same type
point of category theory. A homomorphism may also be an isomorphism, an endomorphism, an automorphism, etc. (see below). Each of those can be defined in a
Homomorphism
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
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
Set whose pairs have minima and maxima
a lattice endomorphism is a lattice homomorphism from a lattice to itself, and a lattice automorphism is a bijective lattice endomorphism. Lattices and
Lattice_(order)
Vector space equipped with a bilinear product
In mathematics, an algebra over a field (often simply called an algebra) is a vector space equipped with a bilinear product. Thus, an algebra is an algebraic
Algebra_over_a_field
In mathematics, invariant of square matrices
determinant of a linear endomorphism determines how the orientation and the n-dimensional volume are transformed under the endomorphism. This is used in calculus
Determinant
Mathematics term
constructed from a particular class of endomorphism of a free monoid. Every automatic sequence is morphic. Let f be an endomorphism of the free monoid A∗ on an alphabet
Morphic_word
Algebraic structure with addition and multiplication
all R-linear maps forms a ring, also called the endomorphism ring and denoted by EndR(V). The endomorphism ring of an elliptic curve. It is a commutative
Ring_(mathematics)
Mathematical term
homomorphism that sends an invertible n-by-n matrix g {\displaystyle g} to an endomorphism of the vector space of all linear transformations of R n {\displaystyle
Adjoint_representation
Mathematical concept in algebra
In linear algebra, a nilpotent matrix is a square matrix N such that N k = 0 {\displaystyle N^{k}=0\,} for some positive integer k {\displaystyle k} .
Nilpotent_matrix
In mathematics, element that equals its square
E. In the case when M = R (assumed unital), the endomorphism ring EndR(R) = R, where each endomorphism arises as left multiplication by a fixed ring element
Idempotent_(ring_theory)
Polynomial whose roots are the eigenvalues of a matrix
polynomial of an endomorphism of a finite-dimensional vector space is the characteristic polynomial of the matrix of that endomorphism over any basis (that
Characteristic_polynomial
Differential mapping
\sigma (x):=x^{p}+p\delta (x)} defines a ring endomorphism which is a lift of the Frobenius endomorphism. When the ring R is p-torsion free the correspondence
P-derivation
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
Sum of elements on the main diagonal
be given using the canonical isomorphism between the space of linear endomorphisms of V of finite rank and V ⊗ V*, where V* is the dual space of V. Let
Trace_(linear_algebra)
Linear map over a ring
homomorphisms. A module homomorphism from a module M to itself is called an endomorphism and an isomorphism from M to itself an automorphism. One writes End R
Module_homomorphism
Defines a notion of parallel transport on a bundle
another by an endomorphism-valued one-form. From this perspective, the connection one-form A {\displaystyle A} is precisely the endomorphism-valued one-form
Connection_(vector_bundle)
Expresses the number of points of a variety over a finite field
endomorphism on its cohomology groups. There are several generalizations: the Frobenius endomorphism can be replaced by a more general endomorphism,
Grothendieck_trace_formula
Linear operator
decomposition expresses an endomorphism x : V → V {\displaystyle x:V\to V} as a sum of a semisimple endomorphism s and a nilpotent endomorphism n such that both
Semisimple_operator
Topic in abstract algebra
modules and associated tilting functors. Here, the second algebra is the endomorphism algebra of a tilting module over the first algebra. Tilting theory was
Tilting_theory
Coordinate change in linear algebra
square matrix of an endomorphism of V on an "old" basis, and P is a change-of-basis matrix, then the matrix of the endomorphism on the "new" basis is
Change_of_basis
Mathematical concept
those for which the endomorphism ring has the maximal possible rank 2. In positive characteristic it is possible for the endomorphism ring to be even larger:
Supersingular_elliptic_curve
Mathematical parametrization of vector spaces by another space
F) over X. Building on the previous example, given a section s of an endomorphism bundle Hom(E, E) and a function f: X → R, one can construct an eigenbundle
Vector_bundle
Homomorphisms between simple modules over the same ring are isomorphisms or zero
of the endomorphism ring of M {\displaystyle M} . Theorem (Lam 2001, §19): A module is said to be strongly indecomposable if its endomorphism ring is
Schur's_lemma
Direct sum of irreducible modules
ring, and every semiprimitive ring is isomorphic to such an image. The endomorphism ring of a semisimple module is not only semiprimitive, but also von Neumann
Semisimple_module
Map (arrow) between two objects of a category
identical source and target) is an endomorphism of X {\displaystyle X} . A split endomorphism is an idempotent endomorphism f {\displaystyle f} if f {\displaystyle
Morphism
Theorem in category theory
A} to the exponential object B A {\displaystyle B^{A}} , then every endomorphism g : B → B {\displaystyle g:B\rightarrow B} has a fixed point. That is
Lawvere's_fixed-point_theorem
Mathematical function between groups that preserves multiplication structure
the set End(G) of all endomorphisms of an abelian group forms a ring, the endomorphism ring of G. For example, the endomorphism ring of the abelian group
Group_homomorphism
Taxonomy to categorize human physiques
ranging from 1 to 7 for each of the three somatotypes, where the pure endomorph is 7–1–1, the pure mesomorph 1–7–1 and the pure ectomorph scores 1–1–7
Somatotype and constitutional psychology
Somatotype_and_constitutional_psychology
Abstract algebra concept
modules. A decomposition with local endomorphism rings (cf. #Azumaya's theorem): a direct sum of modules whose endomorphism rings are local rings (a ring is
Decomposition_of_a_module
From an exceptional automorphism of a Dynkin diagram
an endomorphism whose square is the endomorphism αφ associated to the Frobenius endomorphism φ of the field F. Roughly speaking, this endomorphism απ
Ree_group
Subgroup of a group in mathematics
{\displaystyle g\in G} . The endomorphism σ {\displaystyle \sigma } is an idempotent element in the transformation monoid of endomorphisms, so it is called an
Retract_(group_theory)
the defect, or distance, from being depth two in a tower of iterated endomorphism rings above the subring. A more recent definition of depth of any unital
Depth of noncommutative subrings
Depth_of_noncommutative_subrings
Topics referred to by the same term
End (graph theory) End (group theory) (a subcase of the previous) End (endomorphism) End (gridiron football) End, a division of play in the sports of curling
End
Idempotent linear transformation from a vector space to itself
transformation P {\displaystyle P} from a vector space to itself (an endomorphism) such that P ∘ P = P {\displaystyle P\circ P=P} . That is, whenever P
Projection_(linear_algebra)
Branch of mathematics
\mu _{f}} . A Lattès map is an endomorphism f of C P n {\displaystyle \mathbf {CP} ^{n}} obtained from an endomorphism of an abelian variety by dividing
Complex_dynamics
Generalization of vector spaces from fields to rings
module), and is 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
Module_(mathematics)
Algebraic structure
The skew-polynomial ring is defined similarly for a ring R and a ring endomorphism f of R, by extending the multiplication from the relation X⋅r = f(r)⋅X
Polynomial_ring
Submodule of a mathematical ring
algebras Ring homomorphisms • Kernel • Inner automorphism • Frobenius endomorphism Algebraic structures • Module • Associative algebra • Graded ring • Involutive
Ideal_(ring_theory)
Universal C*-algebra
{\displaystyle b\in A} . A unital *-endomorphism ρ : A → A {\displaystyle \rho :A\to A} is the direct sum of endomorphisms σ 1 , σ 2 , . . . , σ n {\displaystyle
Cuntz_algebra
Mathematical formal group law
unique (1-dimensional) formal group law F such that e(x) = px + xp is an endomorphism of F, in other words e ( F ( x , y ) ) = F ( e ( x ) , e ( y ) ) .
Lubin–Tate_formal_group_law
Subgroup mapped to itself under every automorphism of the parent group
under surjective endomorphisms. For finite groups, surjectivity of an endomorphism implies injectivity, so a surjective endomorphism is an automorphism;
Characteristic_subgroup
Algebraic structure where all polynomials have roots
F is algebraically closed, every endomorphism of Fn has some eigenvector. On the other hand, if every endomorphism of Fn has an eigenvector, let p(x)
Algebraically_closed_field
Matrix whose only nonzero elements are on its main diagonal
This is true more generally for a module M over a ring R, with the endomorphism algebra End(M) (algebra of linear operators on M) replacing the algebra
Diagonal_matrix
In mathematics, the Frobenius endomorphism is defined in any commutative ring R that has characteristic p, where p is a prime number. Namely, the mapping
Arithmetic and geometric Frobenius
Arithmetic_and_geometric_Frobenius
Kind of infinitely long sequence of characters
are Sturmian, and the Sturmian endomorphisms of B∗ are precisely those endomorphisms in the submonoid of the endomorphism monoid generated by {I,φ,ψ}. A
Sturmian_word
conjecture, asked by Jacques Dixmier in 1968, is the conjecture that any endomorphism of a Weyl algebra is an automorphism. Tsuchimoto in 2005, and independently
Dixmier_conjecture
Algebraic structure
are endomorphisms of a medial magma, then the mapping f • g defined by pointwise multiplication (f • g)(x) = f(x) • g(x) is itself an endomorphism. It
Medial_magma
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 group that can be generated as the set of powers of a single element
graphs whose symmetry group includes a transitive cyclic group. The endomorphism ring of the abelian group Z/nZ is isomorphic to Z/nZ itself as a ring
Cyclic_group
Nilpotent subalgebra of a Lie algebra
maximal abelian subalgebra consisting of elements x such that the adjoint endomorphism ad ( x ) : g → g {\displaystyle \operatorname {ad} (x):{\mathfrak {g}}\to
Cartan_subalgebra
Theorem in Lie representation theory
Y ] {\displaystyle \operatorname {ad} (X)(Y)=[X,Y]} , is a nilpotent endomorphism on g {\displaystyle {\mathfrak {g}}} ; i.e., ad ( X ) k = 0 {\displaystyle
Engel's_theorem
Algebraic ring that need not have additive negative elements
suitable multiplication operation arises as the function composition of endomorphisms over any commutative monoid. Some authors define semirings without the
Semiring
x,\,Jy\rangle ,} where the metric operator J {\displaystyle J} is an endomorphism of K {\displaystyle K} obeying J 3 = J . {\displaystyle J^{3}=J.\,} The
Indefinite inner product space
Indefinite_inner_product_space
Ring that is also a vector space or a module
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 an R-algebra by defining (r·φ)(x) = r·φ(x)
Associative_algebra
In algebra, Quillen's lemma states that an endomorphism of a simple module over the enveloping algebra of a finite-dimensional Lie algebra over a field
Quillen's_lemma
Used to compare mixed characteristic situations with purely finite characteristic ones
induced by a nondiscrete valuation of rank 1, such that the Frobenius endomorphism Φ is surjective on K°/p where K° denotes the ring of power-bounded elements
Perfectoid_space
Polynomial associated with a matrix
polynomial always divides some power of the minimal polynomial. Given an endomorphism T on a finite-dimensional vector space V over a field F, let IT be the
Minimal polynomial (linear algebra)
Minimal_polynomial_(linear_algebra)
Algebraic structure
{\displaystyle K} has characteristic p > 0 {\displaystyle p>0} , the Frobenius endomorphism x ↦ x p {\displaystyle x\mapsto x^{p}} is an automorphism. The separable
Perfect_field
Module over the non-commutative Dieudonné ring
of k {\displaystyle k} , and has an endomorphism σ {\displaystyle \sigma } induced by the Frobenius endomorphism of k {\displaystyle k} , so ( w 1 , w
Dieudonné_module
Theorem in topology
the function is [ 0 , 2 ] {\displaystyle [0,2]} . Thus, f is not an endomorphism. Consider the function f ( x ) = x + 1 , {\displaystyle f(x)=x+1,} which
Brouwer_fixed-point_theorem
General theory of mathematical structures
and g ∘ f = 1a. an endomorphism if a = b. end(a) denotes the class of endomorphisms of a. an automorphism if f is both an endomorphism and an isomorphism
Category_theory
Concept in mathematics
software framework. An endomorphism of A∗ is a morphism from A∗ to itself. The identity map I is an endomorphism of A∗, and the endomorphisms form a monoid under
Free_monoid
direct sum of objects having local endomorphism rings. Equivalently, C has split idempotents and the endomorphism ring of every object is semiperfect
Krull–Schmidt_category
Mathematical equivalence relation
similar). That notion corresponds to matrices representing the same endomorphism V → V under two different choices of a single basis of V, used both for
Matrix_equivalence
Algebra over a field where binary multiplication is not necessarily associative
can consider the associative algebra EndK(A) of K-linear vector space endomorphism of A. We can associate to the algebra structure on A two subalgebras
Non-associative_algebra
Method for dividing a simplicial complex
simplicial homology groups, barycentric subdivision can be interpreted as an endomorphism of singular chain complexes. Here again, there exists a subdivision operator
Barycentric_subdivision
length, then every endomorphism of M is either an automorphism or nilpotent. As an immediate consequence, we see that the endomorphism ring of every finite-length
Fitting_lemma
In mathematics, an object whose endomorphisms are isomorphic to another structure
In mathematics, an object whose endomorphisms are isomorphic to another structure
Representation_(mathematics)
Euclidean space without distance and angles
{\displaystyle {\overrightarrow {f}}} . An affine transformation or endomorphism of an affine space A {\displaystyle A} is an affine map from that space
Affine_space
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
Result on the topology of operators on an infinite-dimensional, complex Hilbert space
Hilbert space H. It states that the space GL(H) of invertible bounded endomorphisms of H is such that all maps from any finite complex Y to GL(H) are homotopic
Kuiper's_theorem
Mathematics theory
p-adic curves, the analogue of complex conjugation is the Frobenius endomorphism, and the analogue of the quasi-Fuchsian condition is an integrality condition
P-adic_Teichmüller_theory
Square matrices satisfy their characteristic equation
such endomorphisms. Then φ ∈ End(V) is a possible matrix entry, while A designates the element of M(n, End(V)) whose i, j entry is endomorphism of scalar
Cayley–Hamilton_theorem
Subset of a ring that forms a ring itself
algebras Ring homomorphisms • Kernel • Inner automorphism • Frobenius endomorphism Algebraic structures • Module • Associative algebra • Graded ring • Involutive
Subring
Japanese mathematician (born 1951)
He 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
Not-necessarily-associative commutative algebra satisfying (xy)(xx) = x(y(xx))
exceptional Jordan algebras. A derivation of a Jordan algebra A is an endomorphism D of A such that D(xy) = D(x)y+xD(y). The derivations form a Lie algebra
Jordan_algebra
Algebraic structure with "nice" duality properties
finite-dimensional 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
Frobenius_algebra
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 some
Preadditive_category
(Mathematical) ring with a unique maximal ideal
naturally as endomorphism rings in the study of direct sum decompositions of modules over some other rings. Specifically, if the endomorphism ring of the
Local_ring
Mathematical fallacy
demonstrates that exponentiation by p produces an endomorphism, known as the Frobenius endomorphism of the ring. The demand that the characteristic p
Freshman's_dream
to 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
Invertible linear endomorphism
Yang–Baxter operators are invertible linear endomorphisms with applications in theoretical physics and topology. They are named after theoretical physicists
Yang–Baxter_operator
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
Topics referred to by the same term
circuit boards Ockham algebra, bounded distributive lattice with a dual endomorphism Ockham Awards, annual awards by The Skeptic magazine Ockham New Zealand
Ockham
⊗ O = E for any E. Example: E ⊗ E∗ is canonically isomorphic to the endomorphism bundle End(E), where E∗ is the dual bundle of E. Example: A line bundle
Tensor_product_bundle
Concept in mathematics regarding sets operating on groups
} , the map g ↦ g ω {\displaystyle g\mapsto g^{\omega }} is then an endomorphism of G. From this, it results that a Ω-group can also be viewed as a group
Group_with_operators
Universal representation of a group in terms of its own multiplication
representation for which all elements of the algebra act as the zero linear map (endomorphism) which sends every element of V to the zero vector. For any group or
Trivial_representation
Generalization of associativity properties
We can then define endomorphism operads in this category, as follows. Let V be a finite-dimensional vector space The endomorphism operad E n d V = { E
Operad
Number in {..., –2, –1, 0, 1, 2, ...}
algebras Ring homomorphisms • Kernel • Inner automorphism • Frobenius endomorphism Algebraic structures • Module • Associative algebra • Graded ring • Involutive
Integer
Type of algebraic field extension
(where F is assumed to have prime characteristic p). If the Frobenius endomorphism x ↦ x p {\displaystyle x\mapsto x^{p}} of F is not surjective, there
Separable_extension
Theorem in algebra mathematics
{\displaystyle R} -module and f : M → M {\displaystyle f:M\to M} is a surjective endomorphism, then f {\displaystyle f} is an isomorphism. Over a local ring, one can
Nakayama's_lemma
Mathematical ring whose elements are matrices
of endomorphisms of the free right R-module of rank n; that is, Mn(R) ≅ EndR(Rn). Matrix multiplication corresponds to composition of endomorphisms. The
Matrix_ring
Property of operations
the power set of a monoid to itself are idempotent; the idempotent endomorphisms of a vector space are its projections. If the set E {\displaystyle E}
Idempotence
\ldots ,n\}} This concept 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
Finite_topology
ENDOMORPHISM
ENDOMORPHISM
ENDOMORPHISM
ENDOMORPHISM
Boy/Male
Arthurian Legend
A knight.
Girl/Female
Arabic, Australian, Muslim
Breeze
Boy/Male
English
From the Old English Aethelstan meaning noble stone. Atheistan was a 10th century Anglo-Saxon...
Boy/Male
Muslim
Oath
Girl/Female
Muslim
Praiser
Girl/Female
Hindu, Indian
Gold
Girl/Female
Indian
Flower Name in Sanskrit
Girl/Female
Tamil
Date
Girl/Female
Spanish
Heart.
Girl/Female
Tamil
Smile
ENDOMORPHISM
ENDOMORPHISM
ENDOMORPHISM
ENDOMORPHISM
ENDOMORPHISM