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Andrei Roiter
Andrei Vladimirovich Roiter (''Russian'': Андрей Владимирович Ройтер; ''Ukrainian'': Андрій Володимирович Ройтер, November 30, 1937, Dnipro – July 26, 2006, Riga, Latvia) was a Ukrainian mathematician, specializing in algebra. A. V. Roiter's father was the Ukrainian physical chemist V. A. Roiter, a leading expert on catalysis. In 1955 Andrei V. Roiter matriculated at Taras Shevchenko National University of Kyiv, where he met a fellow mathematics major Lyudmyla Nazarova. In 1958 he and Nazarova transferred to Saint Petersburg State University (then named Leningrad State University). They married and began a lifelong collaboration on representation theory. He received in 1960 his Diploma (M.S.) and in 1963 his Candidate of Sciences degree (PhD). His PhD thesis was supervised by Dmitry Konstantinovich Faddeev, who also supervised Ludmila Nazarova's PhD. A. V. Roiter was hired in 1961 as a researcher at the Institute of Mathematics of t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Dnipro
Dnipro is Ukraine's fourth-largest city, with about one million inhabitants. It is located in the eastern part of Ukraine, southeast of the Ukrainian capital Kyiv on the Dnieper River, Dnipro River, from which it takes its name. Dnipro is the Capital (political), administrative centre of Dnipropetrovsk Oblast. It hosts the administration of Dnipro urban hromada. Dnipro has a population of Archeological evidence suggests the site of the present city was settled by Cossacks, Cossack communities from at least 1524. Yekaterinoslav ("glory of Catherine") was established by decree of the Emperor of all the Russias, Russian Empress Catherine the Great in 1787 as the administrative center of Novorossiya Governorate, Novorossiya. From the end of the 19th century, the town attracted foreign capital and an international, multi-ethnic workforce exploiting Kryvbas iron ore and Donbas coal. Renamed Dnipropetrovsk in 1926 after the Ukrainian Communist Party of the Soviet Union, Communist ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Prime Number
A prime number (or a prime) is a natural number greater than 1 that is not a Product (mathematics), product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime because the only ways of writing it as a product, or , involve 5 itself. However, 4 is composite because it is a product (2 × 2) in which both numbers are smaller than 4. Primes are central in number theory because of the fundamental theorem of arithmetic: every natural number greater than 1 is either a prime itself or can be factorization, factorized as a product of primes that is unique up to their order. The property of being prime is called primality. A simple but slow primality test, method of checking the primality of a given number , called trial division, tests whether is a multiple of any integer between 2 and . Faster algorithms include the Miller–Rabin primality test, which is fast but has a small chance of error ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hereditary Ring
In mathematics, especially in the area of abstract algebra known as module theory, a ring ''R'' is called hereditary if all submodules of projective modules over ''R'' are again projective. If this is required only for finitely generated submodules, it is called semihereditary. For a noncommutative ring ''R'', the terms left hereditary and left semihereditary and their right hand versions are used to distinguish the property on a single side of the ring. To be left (semi-)hereditary, all (finitely generated) submodules of projective ''left'' ''R''-modules must be projective, and similarly to be right (semi-)hereditary all (finitely generated) submodules of projective ''right'' ''R''-modules must be projective. It is possible for a ring to be left (semi-)hereditary but not right (semi-)hereditary and vice versa. Equivalent definitions * The ring ''R'' is left (semi-)hereditary if and only if all ( finitely generated) left ideals of ''R'' are projective modules. * The ring ''R'' i ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Finitely Generated Module
In mathematics, a finitely generated module is a module that has a finite generating set. A finitely generated module over a ring ''R'' may also be called a finite ''R''-module, finite over ''R'', or a module of finite type. Related concepts include finitely cogenerated modules, finitely presented modules, finitely related modules and coherent modules all of which are defined below. Over a Noetherian ring the concepts of finitely generated, finitely presented and coherent modules coincide. A finitely generated module over a field is simply a finite-dimensional vector space, and a finitely generated module over the integers is simply a finitely generated abelian group. Definition The left ''R''-module ''M'' is finitely generated if there exist ''a''1, ''a''2, ..., ''a''''n'' in ''M'' such that for any ''x'' in ''M'', there exist ''r''1, ''r''2, ..., ''r''''n'' in ''R'' with ''x'' = ''r''1''a''1 + ''r''2''a''2 + ... + ''r''''n''''a''''n''. The set is referred to as a gene ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Category (mathematics)
In mathematics, a category (sometimes called an abstract category to distinguish it from a concrete category) is a collection of "objects" that are linked by "arrows". A category has two basic properties: the ability to compose the arrows associatively and the existence of an identity arrow for each object. A simple example is the category of sets, whose objects are sets and whose arrows are functions. ''Category theory'' is a branch of mathematics that seeks to generalize all of mathematics in terms of categories, independent of what their objects and arrows represent. Virtually every branch of modern mathematics can be described in terms of categories, and doing so often reveals deep insights and similarities between seemingly different areas of mathematics. As such, category theory provides an alternative foundation for mathematics to set theory and other proposed axiomatic foundations. In general, the objects and arrows may be abstract entities of any kind, and the n ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Subcategory
In mathematics, specifically category theory, a subcategory of a category ''C'' is a category ''S'' whose objects are objects in ''C'' and whose morphisms are morphisms in ''C'' with the same identities and composition of morphisms. Intuitively, a subcategory of ''C'' is a category obtained from ''C'' by "removing" some of its objects and arrows. Formal definition Let ''C'' be a category. A subcategory ''S'' of ''C'' is given by *a subcollection of objects of ''C'', denoted ob(''S''), *a subcollection of morphisms of ''C'', denoted hom(''S''). such that *for every ''X'' in ob(''S''), the identity morphism id''X'' is in hom(''S''), *for every morphism ''f'' : ''X'' → ''Y'' in hom(''S''), both the source ''X'' and the target ''Y'' are in ob(''S''), *for every pair of morphisms ''f'' and ''g'' in hom(''S'') the composite ''f'' o ''g'' is in hom(''S'') whenever it is defined. These conditions ensure that ''S'' is a category in its own right: its collection of objects is ob(''S'' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Auslander–Reiten Theory
In algebra, Auslander–Reiten theory studies the representation theory of Artinian rings using techniques such as Auslander–Reiten sequences (also called almost split sequences) and Auslander–Reiten quivers. Auslander–Reiten theory was introduced by and developed by them in several subsequent papers. For survey articles on Auslander–Reiten theory see , , , and the book . Many of the original papers on Auslander–Reiten theory are reprinted in . Almost-split sequences Suppose that ''R'' is an Artin algebra. A sequence :0→ ''A'' → ''B'' → ''C'' → 0 of finitely generated left modules over ''R'' is called an almost-split sequence (or Auslander–Reiten sequence) if it has the following properties: *The sequence is not split *''C'' is indecomposable and any homomorphism from an indecomposable module to ''C'' that is not an isomorphism factors through ''B''. *''A'' is indecomposable and any homomorphism from ''A'' to an indecomposable module that is not a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Maurice Auslander
Maurice Auslander (August 3, 1926 – November 18, 1994) was an American mathematician who worked on commutative algebra, homological algebra and the representation theory of Artin algebras (e.g. finite-dimensional associative algebras over a field). He proved the Auslander–Buchsbaum theorem that regular local rings are factorial, the Auslander–Buchsbaum formula, and, in collaboration with Idun Reiten, introduced Auslander–Reiten theory and Auslander algebras. Born in Brooklyn, New York, Auslander received his bachelor's degree and his Ph.D. (1954) from Columbia University. He was a visiting scholar at the Institute for Advanced Study in 1956-57. He was a professor at Brandeis University from 1957 until his death in Trondheim, Norway aged 68. He was elected a Fellow of the American Academy of Arts and Sciences in 1971. Upon his death he was survived by his mother, his widow, a daughter, and a son. His widow Bernice L. Auslander (November 21, 1931 - June 18, 2022) wa ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Artin Algebra
In algebra, an Artin algebra is an algebra Λ over a commutative Artin ring ''R'' that is a finitely generated ''R''-module. They are named after Emil Artin. Every Artin algebra is an Artin ring. Dual and transpose There are several different dualities taking finitely generated modules over Λ to modules over the opposite algebra Λop. *If ''M'' is a left Λ-module then the right Λ-module ''M''* is defined to be HomΛ(''M'',Λ). * The dual ''D''(''M'') of a left Λ-module ''M'' is the right Λ-module ''D''(''M'') = Hom''R''(''M'',''J''), where ''J'' is the dualizing module of ''R'', equal to the sum of the injective envelopes of the non-isomorphic simple ''R''-modules or equivalently the injective envelope of ''R''/rad ''R''. The dual of a left module over Λ does not depend on the choice of ''R'' (up to isomorphism). *The transpose Tr(''M'') of a left Λ-module ''M'' is a right Λ-module defined to be the cokernel of the map ''Q''* → ''P''*, where '' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Algebra Over A Field
In mathematics, an algebra over a field (often simply called an algebra) is a vector space equipped with a bilinear map, bilinear product (mathematics), product. Thus, an algebra is an algebraic structure consisting of a set (mathematics), set together with operations of multiplication and addition and scalar multiplication by elements of a field (mathematics), field and satisfying the axioms implied by "vector space" and "bilinear". The multiplication operation in an algebra may or may not be associative, leading to the notions of associative algebras where associativity of multiplication is assumed, and non-associative algebras, where associativity is not assumed (but not excluded, either). Given an integer ''n'', the ring (mathematics), ring of real matrix, real square matrix, square matrices of order ''n'' is an example of an associative algebra over the field of real numbers under matrix addition and matrix multiplication since matrix multiplication is associative. Three-dime ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Dimension (vector Space)
In mathematics, the dimension of a vector space ''V'' is the cardinality (i.e., the number of vectors) of a Basis (linear algebra), basis of ''V'' over its base Field (mathematics), field. p. 44, §2.36 It is sometimes called Hamel dimension (after Georg Hamel) or algebraic dimension to distinguish it from other types of dimension. For every vector space there exists a basis, and all bases of a vector space have equal cardinality; as a result, the dimension of a vector space is uniquely defined. We say V is if the dimension of V is wiktionary:finite, finite, and if its dimension is infinity, infinite. The dimension of the vector space V over the field F can be written as \dim_F(V) or as [V : F], read "dimension of V over F". When F can be inferred from context, \dim(V) is typically written. Examples The vector space \R^3 has \left\ as a standard basis, and therefore \dim_(\R^3) = 3. More generally, \dim_(\R^n) = n, and even more generally, \dim_(F^n) = n for any Field (mathe ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ring (mathematics)
In mathematics, a ring is an algebraic structure consisting of a set with two binary operations called ''addition'' and ''multiplication'', which obey the same basic laws as addition and multiplication of integers, except that multiplication in a ring does not need to be commutative. Ring elements may be numbers such as integers or complex numbers, but they may also be non-numerical objects such as polynomials, square matrices, functions, and power series. A ''ring'' may be defined as a set that is endowed with two binary operations called ''addition'' and ''multiplication'' such that the ring is an abelian group with respect to the addition operator, and the multiplication operator is associative, is distributive over the addition operation, and has a multiplicative identity element. (Some authors apply the term ''ring'' to a further generalization, often called a '' rng'', that omits the requirement for a multiplicative identity, and instead call the structure defi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
