About the mathematical concept the endomorphic body type Somatotype In mathematics , an endomorphism is a morphism or homomorphism from a mathematical object to itself. For example, an endomorphism of a vector space V is a linear map V V , and an endomorphism of a group mathematics group G is a group homomorphism G G . In general, we can talk about endomorphisms in any category theory category . In the category of Set mathematics sets , endomorphisms are simply functions from a set S into itself. In any category, the function composition composition of any two endomorphisms of X is again an endomorphism of X . It follows that the set of all endomorphisms of X forms a monoid , denoted End X or End sub C sub X to emphasize the category C . An inverse element invertible endomorphism of X is called an automorphism . The set of all automorphisms is a subset of End X with a group mathematics group structure, called the automorphism group of X and denoted Aut X . In the following diagram, the arrows denote implication border 0 align center width 42 automorphism align center width 16 math Rightarrow math align center width 42 isomorphism align center math Downarrow math align center math Downarrow math align center endomorphism align center math Rightarrow ..., the endomorphisms of an abelian group form a ring mathematics ring the endomorphism ring . For example ... generate an algebraic structure known as a nearring . Every ring with one is the endomorphism ring of its regular module , and so is a subring of an endomorphism ring of an abelian group, ref Jacobson 2009 , p. 162, Theorem 3.2. ref however there are rings which are not the endomorphism ring of any ... is an endomorphism. Let S be an arbitrary set. Among endofunctions on S one finds permutation .... Notes references See also morphism Frobenius endomorphism adjoint endomorphism References ... id 7462 title Endomorphism Category Morphisms ca Endomorfisme de Endomorphismus et Endomorfism fr Endomorphisme ... more details
In abstract algebra , the endomorphism ring of an Abelian group X , denoted generically by End X , is a set ... on the category mathematics category of the Abelian group under examination. The endomorphism ring ... theory algebra over some ring R, this may also be called the endomorphism algebra . Description Let ... ring which is not a ring. Examples In the category of R module mathematics module s the endomorphism ... , the endomorphism ring is central to Morita equivalence of module categories. If K is a field mathematics field and we consider the K vector space K sup n sup , then the endomorphism ring of K sup n ... of n by n matrices with entries in K sfn Drozd Kirichenko 1994 loc pp. 23 31 . More generally, the endomorphism ... the elements of R act on R by left multiplication. In general, endomorphism rings can be defined for the objects of any preadditive category . Properties Endomorphism rings always have multiplicative identity element identity , namely the identity function identity map . Endomorphism rings are typically non commutative ring non commutative . If a module is simple module simple , then its endomorphism ... . A module is indecomposable module indecomposable if and only if its endomorphism ring does not contain ... module , then indecomposability is equivalent to the endomorphism ring being a local ring sfn Wisbauer 1991 loc p.163 . For a semisimple module , the endomorphism ring is a von Neumann regular ring . The endomorphism ring of a nonzero right uniserial module has either one or two maximal right ideals. If the module is Artinian, Noetherian, projective or injective, then the endomorphism ring has a unique maximal ideal, so that it is a local ring. The endomorphism ring of a an Artinian uniform module is a local ring sfn Wisbauer 1991 loc p. 263 . The endomorphism ring of a module with finite composition length is a semiprimary ring . The endomorphism ring of a continuous module or discrete ... and projective that is, a progenerator , then the endomorphism ring of the module and R share ... more details
In commutative algebra and field theory mathematics field theory , the Frobenius endomorphism after Ferdinand Georg Frobenius is a special endomorphism of commutative Ring mathematics rings with prime characteristic algebra characteristic p , an important class which includes finite Field mathematics fields . The endomorphism maps every element to its p th power. In certain contexts it is an automorphism , but this is not true in general. Definition Let R be a commutative ring with prime characteristic p an integral domain of positive characteristic always has prime characteristic, for example . The Frobenius endomorphism F is defined by F r r sup p sup for all r in R . Clearly this respects the multiplication of R F rs rs sup p sup r sup p sup s sup p sup F r F s , and F 1 is clearly 1 also. What is interesting, however, is that it also respects the addition of R . The expression r s sup p sup can be expanded using the binomial theorem . Because p is prime, it divides p but not any q for q p it therefore will divide the numerator , but not the denominator , of the explicit formula of the binomial coefficient s math frac p k p k math for 1 k p &minus 1. Therefore the coefficients of all the terms except r sup p sup and s sup p sup are divisible by p , the characteristic, and hence they vanish. ref This is known as the Freshman s dream . ref Thus F r s r s sup p sup r sup p sup s sup ... is F sub p sub T T sup p sup , where F T 0, but T 0. Fixed points of the Frobenius endomorphism ... . This forces R to be a characteristic p ring, so we can define the Frobenius endomorphism F for R as we did above. It is clear that F commutes with localization, so F glues to give an endomorphism of X , called the absolute Frobenius map . However, F is not necessarily an endomorphism of k schemes ... field s, there is a concept of Frobenius endomorphism which induces the Frobenius endomorphism in the corresponding ... title Frobenius automorphism Springer id f f041770 title Frobenius endomorphism Category Finite fields ... more details
Lie groups In mathematics , the adjoint endomorphism or adjoint action is an endomorphism of Lie algebra s that plays a fundamental role in the development of the theory of Lie algebras and Lie groups . Given an element x of a Lie algebra math mathfrak g math , one defines the adjoint action of x on math mathfrak g math as the endomorphism math textrm ad x mathfrak g to mathfrak g math with math textrm ad x y x,y math for all y in math mathfrak g math . math textrm ad x math is an group action action that is linear . Adjoint representation The mapping math textrm ad mathfrak g rightarrow operatorname Der mathfrak g subset textrm End mathfrak g mathfrak gl mathfrak g math given by math x mapsto textrm ad x math is a representation of a Lie algebra and is called the adjoint representation of the algebra. Here, math mathfrak gl mathfrak g math is the Lie algebra of the general linear group over the vector space math mathfrak g math . It is isomorphic to math textrm End mathfrak g math . Within math mathfrak gl mathfrak g math , the composition of two maps is well defined, and the Lie bracket of vector fields Lie bracket may be shown to be given by the commutator of the two elements, math textrm ad x, textrm ad y textrm ad x circ textrm ad y textrm ad y circ textrm ad x math where math circ math denotes composition of linear maps. If math mathfrak g math is finite dimensional and a basis for it is chosen, this corresponds to matrix multiplication . Using this and the definition of the Lie bracket in terms of the mapping ad above, the Jacobi identity math x, y,z y, z,x z, x,y 0 math takes the form math left textrm ad x, textrm ad y right z left textrm ad x,y right z math where x , y , and z are arbitrary elements of math mathfrak g math . This last identity confirms that ad really is a Lie algebra homomorphism, in that the morphism ad commutes with the multiplication operator , . The kernel of math operatorname ad mathfrak g to operatorname ad mathfrak g math is, by definition ... more details
Endomorph , endomorphic , and endomorphism can refer to One of the three somatotype s, or animal body types, that contains high body fat, and that gains weight easily the other two are ectomorphic and mesomorphic Endomorphism can also refer to a mathematical concept In category theory, something pertaining to or related by an endomorphism disambig ... more details
The adjoint representation can refer to Ad the adjoint representation of a Lie group G ad the adjoint representation of a Lie algebra math mathfrak g math , see adjoint endomorphism . mathdab ... more details
In mathematics , in the field of group theory , a subgroup of a group mathematics group is termed a retract if there is an endomorphism of the group that maps surjective ly to the subgroup and is identity on the subgroup. In symbols, math H math is a retract of math G math if and only if there is an endomorphism math sigma G to G math such that math sigma h h math for all math h in H math and math sigma g in H math for all math g in G math . The endomorphism itself is termed an idempotent endomorphism or a retraction. The following is known about retracts A subgroup is a retract if and only if it has a normal subgroup normal complement group theory complement . The normal complement, specifically, is the kernel of the retraction. Every direct product of groups direct factor is a retract. Conversely, any retract which is a normal subgroup is a direct factor. Every retract has the CEP subgroup congruence extension property . Every regular factor , and in particular, every free factor , is a retract. References unreferenced date September 2008 Category Group theory Category Subgroup properties Abstract algebra stub ... more details
of R modules is either invertible or zero. In particular, the endomorphism ring of a simple module ... that the endomorphism ring of the module M is a division ring this division ring contains C in its center, is finite dimensional over C and is therefore equal to C . Thus the endomorphism ring of the module ... closed, the case where the endomorphism ring is as small as possible is of particular ... if its endomorphism ring is isomorphic to k . This is in general stronger than being irreducible over ... of the endomorphism ring of M . A module is said to be strongly indecomposable if its endomorphism ... module indecomposable M is strongly indecomposable Every endomorphism of M is either nilpotent or invertible ... their endomorphism algebra is a division ring . Such modules are necessarily indecomposable, and so ... over the ring of integer s, the module of rational number s has an endomorphism ring that is a division ... with three elements has the field with three elements as its endomorphism ring. In general, multiplicity ... more details
Unreferenced date March 2009 In mathematics , the medial Category mathematics category Med , that is, the category of medial Magma algebra magma s has as objects sets with a medial binary operation , and morphism s given by homomorphism s of operations in the universal algebra sense . The category Med has Direct product Categorical product direct product s, so the concept of a medial magma object internal binary operation makes sense. As a result, Med has all its objects as medial objects , and this characterizes it. There is an inclusion functor from Set to Med as trivial magma algebra magma s, with binary operation operation s being the right projection mathematics projection s x , y &rarr y . An injective endomorphism can be extended to an automorphism of a magma extension algebra extension &mdash the colimit of the constant sequence of the endomorphism . See also Eckmann Hilton argument Category Category theoretic categories Medial magmas es Medial categor a ... more details
In mathematics , a subgroup of a group mathematics group is fully characteristic or fully invariant if it is Invariant mathematics invariant under every endomorphism of the group. That is, any endomorphism of the group takes elements of the subgroup to elements of the subgroup. Every group has itself the improper subgroup and the trivial subgroup as two of its fully characteristic subgroups. Every fully characteristic subgroup is a strictly characteristic subgroup , and a fortiori a characteristic subgroup . The commutator subgroup of a group is always a fully characteristic subgroup. More generally, any verbal subgroup is always fully characteristic. For any reduced free group , and, in particular, for any free group , the converse also holds &mdash every fully characteristic subgroup is verbal. See also characteristic subgroup . References cite book title Group Theory first W.R. last Scott pages 45 46 publisher Dover year 1987 isbn 0 486 65377 3 cite book title Combinatorial Group Theory first Wilhelm last Magnus coauthors Abraham Karrass, Donald Solitar publisher Dover year 2004 pages 74 85 isbn 0 486 43830 9 Category Subgroup properties ... more details
In mathematics , the Frobenius endomorphism is defined in any commutative ring R that has characteristic algebra characteristic p , where p is a prime number . Namely, the mapping that takes r in R to r sup p sup is a ring endomorphism of R . The image of is then R sup p sup , the subring of R consisting of p th powers. In some important cases, for example finite field s, is surjective . Otherwise is an endomorphism but not a ring automorphism . The terminology of geometric Frobenius arises by applying the spectrum of a ring construction to . This gives a mapping Spec R sup p sup Spec R of affine scheme s. Even in cases where R sup p sup R this is not the identity, unless R is the prime field . Mappings created by fibre product with , i.e. Grothendieck s relative point of view base change s, tend in scheme theory to be called geometric Frobenius . The reason for a careful terminology is that the Frobenius automorphism in Galois group s, or defined by transport of structure , is often the inverse mapping of the geometric Frobenius. As in the case of a cyclic group in which a generator is also the inverse of a generator, there are in many situations two possible definitions of Frobenius, and without a consistent convention some problem of a minus sign may appear. References Citation last1 Freitag first1 Eberhard last2 Kiehl first2 Reinhardt title tale cohomology and the Weil conjecture publisher Springer Verlag location Berlin, New York series Ergebnisse der Mathematik und ihrer Grenzgebiete 3 Results in Mathematics and Related Areas 3 isbn 978 3 540 12175 6 mr 926276 year 1988 volume 13 , p. 5 DEFAULTSORT Arithmetic And Geometric Frobenius Category Mathematical terminology Category Algebraic geometry Category Algebraic number theory ... more details
refimprove date March 2012 In mathematics , the category mathematics category of magma algebra magmas , denoted Mag , has as objects sets with a binary operation , and morphism s given by homomorphism s of operations in the universal algebra sense . The category Mag has product category theory direct products , so the concept of a magma object internal binary operation clarification needed date March 2012 makes sense. As in any category with direct products . There is an inclusion functor nobr category of sets Set category of medial magmas Med Mag as trivial magma algebra magma s, with binary operation operation s given by projection mathematics projection math 1 x T y y . An important property is that an injective endomorphism can be extended to an automorphism of a magma algebraic extension extension , just the colimit of the constant function constant sequence of the endomorphism . Because the Singleton mathematics singleton math , is the zero object of Mag , fact date March 2012 it is terminal, obviously. but who said that the empty magma is banned? and because Mag is algebraic category algebraic , Mag is pointed and complete category complete . ref Cite book title Mal cev, protomodular, homological and semi abelian categories first Francis last Borceux first2 Dominique last2 Bourn publisher Springer year 2004 pages 7, 19 isbn 1402019610 ref References references DEFAULTSORT Category Of Magmas Category Category theoretic categories Magmas es Magma categor a nl Categorie van magma s ... more details
In mathematics , the Eisenstein ideal is a certain ideal ring theory ideal in the endomorphism ring of the Jacobian variety of a modular curve . It was introduced by Barry Mazur in 1977, in studying the rational points of modular curves. The endomorphism ring in question is closely associated with a Hecke algebra , and the name comes from the way the definition in detail follows the action of Hecke operator s on Eisenstein series . Let N be a positive integer , and define J sub 0 sub N J as the Jacobian variety of the modular curve X sub 0 sub N X . There are endomorphisms T sub l sub of J for each prime number l not dividing N . These come from the Hecke operator, considered first as an Correspondence mathematics algebraic correspondence on X , and from there as acting on divisor class es, which gives the action on J . There is also an involution w . The Eisenstein ideal, in the unital subring of End J generated as a ring by the T sub l sub , is generated as an ideal by the elements T sub l sub &minus l 1 for all l not dividing N , and by w 1. References Mazur, B. Modular curves and the Eisenstein ideal. Inst. Hautes tudes Sci. Publ. Math. No. 47 1977 , 33 186 1978 . Category Modular forms Category Abelian varieties ... more details
In mathematics, the term adjoint applies in several situations. Several of these share a similar formalism if A is adjoint to B , then there is typically some formula of the type Ax , y x , By . Specifically, adjoint may mean Hermitian adjoint adjoint of a linear operator in functional analysis Adjoint functors in category theory Adjoint representation of a Lie group Adjoint endomorphism of a Lie algebra Conjugate transpose of a matrix in linear algebra Adjugate matrix , related to its inverse Adjoint equation The upper and lower adjoints of a Galois connection in order theory For the adjoint of a differential operator with general polynomial coefficients see differential operator Operator methods differential operator Category Mathematical terminology ... more details
In mathematics , in the realm of Abelian group abelian group theory , a Group mathematics group is said to be algebraically compact if it is a direct summand of every abelian group containing it as a pure subgroup . Equivalent characterizations of algebraic compactness The group is complete in the math mathbb Z math adic topology. The group is pure injective , that is, injective with respect to exact sequences where the embedding is as a pure subgroup. Relations with other properties A torsion free group is cotorsion group cotorsion if and only if it is algebraically compact. Every injective group is algebraically compact. Ulm factor s of cotorsion groups are algebraically compact. External links http www.springerlink.com index W3W06361813J347X.pdf On endomorphism rings of Abelian groups Category Abelian group theory Category Properties of groups ... more details
DISPLAYTITLE p derivation In mathematics , more specifically differential algebra , a p derivation for p a prime number on a Ring mathematics ring R , is a mapping from R to R that satisfies certain conditions outlined directly below. The notion of a p derivation is related to that of a derivation in differential algebra. Definition Let p be a prime number. A p derivation on a ring math R math is a map of sets math delta R to R math that satisfies the following product rule math delta ab delta a b p a p delta b p delta a delta b math and sum rule math delta a b delta a delta b frac a p b p a b p p math . Note that in the sum rule we are not really dividing by p , since all the relevant binomial coefficients in the numerator are divisible by p , so this definition applies in the case when math R math has p Torsion algebra torsion . Relation to Frobenius Endomorphisms A map math sigma R to R math is a lift of the Frobenius endomorphism provided math sigma x x p mod pR math . An example such lift could come from the Artin map . If math R, delta math is a ring with a p derivation, then the map math sigma x x p p delta x math defines a ring endomorphism which is a lift of the frobenius endomorphism. When the ring R is p torsion free the correspondence is a bijection. Examples For math R mathbb Z math the unique p derivation is the map math delta x frac x x p p . math The quotient is well defined because of Fermat s Little Theorem . If R is any p torsion free ring and math sigma R to R math is a lift of the Frobenius endomorphism then math delta x frac sigma x x p p math defines a p derivation. See also Derivation References Citation first Alex last Buium title Arithmetic Differential Equations year 1989 publisher Springer Verlag isbn 0 8218 3862 8 series Mathematical Surveys and Monographs . External links http projecteuclid.org DPubS?verb Display&version 1.0&service UI&handle euclid.dmj 1077245037&page record Project Euclid Category Differential algebra ... more details
In mathematics , the Auslander algebra of an algebra A is the endomorphism ring of the sum of the indecomposable module s of A . It was introduced by harvs txt authorlink Maurice Auslander last Auslander year 1974 . An Artin algebra is called an Auslander algebra if gl dim 2 and if 0 I J K 0 is a minimal injective resolution of then I and J are projective modules References Citation last1 Auslander first1 Maurice title Representation theory of Artin algebras. II doi 10.1080 00927877409412807 mr 0349747 year 1974 journal Communications in Algebra issn 0092 7872 volume 1 issue 4 pages 269 310 Category Representation theory Abstract algebra stub ... more details
In mathematics, a Rosati involution , named after Carlo Rosati , is an involution of the endomorphism ring of an abelian variety induced by a polarization. References Citation last1 Mumford first1 David author1 link David Mumford title Abelian varieties origyear 1970 publisher American Mathematical Society location Providence, R.I. series Tata Institute of Fundamental Research Studies in Mathematics isbn 978 81 85931 86 9 oclc 138290 id MR 0282985 year 2008 volume 5 Citation last1 Rosati first1 Carlo title Sulle corrispondenze algebriche fra i punti di due curve algebriche. language Italian doi 10.1007 BF02419717 year 1918 journal Annali di Matematica Pura ed Applicata volume 3 issue 28 pages 35 60 Category Algebraic geometry ... more details
Frobenius can be Ferdinand Georg Frobenius 1849 1917 , mathematician Frobenius algebra Frobenius endomorphism Frobenius inner product Frobenius norm Frobenius method Frobenius group Frobenius theorem differential topology Frobenius Orgelbyggeri , Danish organ building firm Georg Ludwig Frobenius 1566 1645 , German publisher Johann Froben Johannes Frobenius 1460 1527 , publisher and printer in Basel Hieronymus Froben Hieronymus Frobenius 1501 1563 , publisher and printer in Basel, son of Johannes Ambrosius Frobenius 1537 1602 , publisher and printer in Basel, son of Hieronymus Leo Frobenius 1873 1938 , ethnographer Nikolaj Frobenius b. 1965 , Norwegian writer and screenwriter August Sigmund Frobenius 1741 , German chemist disambig Category Surnames es Frobenius fr Frobenius ja ru ... more details
, as is shown by the second example above. By looking at the endomorphism ring of a module, one can tell whether the module is indecomposable if and only if the endomorphism ring does not contain an idempotent different from 0 and 1. ref name Jacobson If f is such an idempotent endomorphism of M , then M ... if and only if its endomorphism ring is local ring local . Still more information about endomorphisms ... more details
In algebra the Dixmier conjecture , asked by harvtxt Dixmier 1968 loc problem 1 , is the conjecture that any endomorphism of a Weyl algebra is an automorphism. harvtxt Belov Kanel Kontsevich 2007 showed that the Dixmier conjecture generalized to Weyl algebras with more generators is equivalent to the Jacobian conjecture . References Citation last1 Dixmier first1 Jacques title Sur les alg bres de Weyl url http www.numdam.org item?id BSMF 1968 96 209 0 mr 0242897 year 1968 journal Bulletin de la Soci t Math matique de France volume 96 pages 209 242 Tsuchimoto, Yoshifumi. Endomorphisms of Weyl algebra and p curvatures . Osaka J. Math. 42 2005 , 435 452. Citation last1 Belov Kanel first1 Alexei last2 Kontsevich first2 Maxim title The Jacobian conjecture is stably equivalent to the Dixmier conjecture arxiv math 0512171 mr 2337879 year 2007 journal Moscow Mathematical Journal volume 7 issue 2 pages 209 218 Category Algebra Category Conjectures algebra stub ... more details
In mathematics , in particular in the field of the group representation representation theory of groups , a representative function is a function mathematics function f on a compact group compact topological group G obtained by Function composition composing a representation of G on a vector space V with a linear map from the endomorphism s of V into V s underlying field mathematics field . Representative functions arise naturally from finite dimensional representations of G as the matrix mathematics matrix entry functions of the corresponding matrix representations. It follows from the Peter Weyl theorem that the representative functions on G are dense in the Hilbert space of square integrable functions on G . References Theodor Br cker and Tammo tom Dieck, Representations of compact Lie groups , Graduate Texts in Mathematics 98 , Springer Verlag, Berlin, 1995. Category Unitary representation theory Category Representation theory of Lie groups Category Types of functions algebra stub ... more details
In algebraic geometry , a Humbert surface , studied by harvs txt last Humbert authorlink Marie Georges Humbert year 1899 , is a surface in the moduli space of principally polarized abelian surface s consisting of the surfaces with a symmetric endomorphism of some fixed discriminant . References Citation last1 Hulek first1 Klaus last2 Kahn first2 Constantin last3 Weintraub first3 Steven H. title Moduli spaces of abelian surfaces compactification, degenerations, and theta functions url http books.google.com books?id oC2ZF2TeaYAC publisher Walter de Gruyter & Co. location Berlin series de Gruyter Expositions in Mathematics isbn 978 3 11 013851 1 id MR 1257185 year 1993 volume 12 Marie Georges Humbert Humbert, G. , Sur les fonctionnes ab liennes singuli res. I, II, III. J. Math. Pures Appl. serie 5, t. V, 233&ndash 350 1899 t. VI, 279&ndash 386 1900 t. VII, 97&ndash 123 1901 Category Algebraic surfaces Category Complex surfaces ... more details
to an endomorphism of an elliptic complex endomorphism of the elliptic complex T . Such a T has its Lefschetz number L T which by definition is the alternating sum of its trace of an endomorphism traces ... sub j , sub at a fixed point x of f , and x is the determinant of the endomorphism I &minus Df at x ... calculating the Lefschetz number of an endomorphism of an elliptic complex. M. F. Atiyah R. Bott ... more details
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