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Wedderburn–Artin Theorem
In algebra, the Wedderburn–Artin theorem is a classification theorem for semisimple rings and semisimple algebras. The theorem states that an (Artinian) semisimple ring ''R'' is isomorphic to a product of finitely many -by- matrix rings over division rings , for some integers , both of which are uniquely determined up to permutation of the index . In particular, any simple left or right Artinian ring is isomorphic to an ''n''-by-''n'' matrix ring over a division ring ''D'', where both ''n'' and ''D'' are uniquely determined. Theorem Let be a (Artinian) semisimple ring. Then the Wedderburn–Artin theorem states that is isomorphic to a product of finitely many -by- matrix rings M_(D_i) over division rings , for some integers , both of which are uniquely determined up to permutation of the index . There is also a version of the Wedderburn–Artin theorem for algebras over a field . If is a finite-dimensional semisimple -algebra, then each in the above statement is ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Algebra
Algebra () is one of the broad areas of mathematics. Roughly speaking, algebra is the study of mathematical symbols and the rules for manipulating these symbols in formulas; it is a unifying thread of almost all of mathematics. Elementary algebra deals with the manipulation of variables (commonly represented by Roman letters) as if they were numbers and is therefore essential in all applications of mathematics. Abstract algebra is the name given, mostly in education, to the study of algebraic structures such as groups, rings, and fields (the term is no more in common use outside educational context). Linear algebra, which deals with linear equations and linear mappings, is used for modern presentations of geometry, and has many practical applications (in weather forecasting, for example). There are many areas of mathematics that belong to algebra, some having "algebra" in their name, such as commutative algebra, and some not, such as Galois theory. The word ''algebra'' is ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Simple Module
In mathematics, specifically in ring theory, the simple modules over a ring ''R'' are the (left or right) modules over ''R'' that are non-zero and have no non-zero proper submodules. Equivalently, a module ''M'' is simple if and only if every cyclic submodule generated by a element of ''M'' equals ''M''. Simple modules form building blocks for the modules of finite length, and they are analogous to the simple groups in group theory. In this article, all modules will be assumed to be right unital modules over a ring ''R''. Examples Z-modules are the same as abelian groups, so a simple Z-module is an abelian group which has no non-zero proper subgroups. These are the cyclic groups of prime order. If ''I'' is a right ideal of ''R'', then ''I'' is simple as a right module if and only if ''I'' is a minimal non-zero right ideal: If ''M'' is a non-zero proper submodule of ''I'', then it is also a right ideal, so ''I'' is not minimal. Conversely, if ''I'' is not minimal, then t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Brauer Group
Brauer or Bräuer is a surname of German origin, meaning "brewer". Notable people with the name include:- * Alfred Brauer (1894–1985), German-American mathematician, brother of Richard * Andreas Brauer (born 1973), German film producer * Arik Brauer (1929–2021), Austrian painter, poet, and actor, father of Timna Brauer * August Brauer (1863-1917), German zoologist * Friedrich Moritz Brauer (1832–1904), Austrian entomologist and museum director * Georg Brauer (1908–2001), German chemist * Ingrid Arndt-Brauer (born 1961), German politician; member of the Bundestag * Jono Brauer (born 1981), Australian Olympic skier * Max Brauer (1887–1973), German politician; First Mayor of Hamburg * Michael Brauer (contemporary), American audio engineer * Rich Brauer (born 1954), American politician from Illinois; state legislator since 2003 * Richard Brauer (1901–1977), German-American mathematician * Richard H. W. Brauer (contemporary), American art museum director; eponym of the Bra ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Maschke's Theorem
In mathematics, Maschke's theorem, named after Heinrich Maschke, is a theorem in group representation theory that concerns the decomposition of representations of a finite group into irreducible pieces. Maschke's theorem allows one to make general conclusions about representations of a finite group ''G'' without actually computing them. It reduces the task of classifying all representations to a more manageable task of classifying irreducible representations, since when the theorem applies, any representation is a direct sum of irreducible pieces (constituents). Moreover, it follows from the Jordan–Hölder theorem that, while the decomposition into a direct sum of irreducible subrepresentations may not be unique, the irreducible pieces have well-defined multiplicities. In particular, a representation of a finite group over a field of characteristic zero is determined up to isomorphism by its character. Formulations Maschke's theorem addresses the question: when is a gener ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Central Simple Algebra
In ring theory and related areas of mathematics a central simple algebra (CSA) over a field ''K'' is a finite-dimensional associative ''K''-algebra ''A'' which is simple, and for which the center is exactly ''K''. (Note that ''not'' every simple algebra is a central simple algebra over its center: for instance, if ''K'' is a field of characteristic 0, then the Weyl algebra K ,\partial_X/math> is a simple algebra with center ''K'', but is ''not'' a central simple algebra over ''K'' as it has infinite dimension as a ''K''-module.) For example, the complex numbers C form a CSA over themselves, but not over the real numbers R (the center of C is all of C, not just R). The quaternions H form a 4-dimensional CSA over R, and in fact represent the only non-trivial element of the Brauer group of the reals (see below). Given two central simple algebras ''A'' ~ ''M''(''n'',''S'') and ''B'' ~ ''M''(''m'',''T'') over the same field ''F'', ''A'' and ''B'' are called ''similar'' (or ''Brauer equ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Algebraically Closed Field
In mathematics, a field is algebraically closed if every non-constant polynomial in (the univariate polynomial ring with coefficients in ) has a root in . Examples As an example, the field of real numbers is not algebraically closed, because the polynomial equation ''x''2 + 1 = 0 has no solution in real numbers, even though all its coefficients (1 and 0) are real. The same argument proves that no subfield of the real field is algebraically closed; in particular, the field of rational numbers is not algebraically closed. Also, no finite field ''F'' is algebraically closed, because if ''a''1, ''a''2, ..., ''an'' are the elements of ''F'', then the polynomial (''x'' − ''a''1)(''x'' − ''a''2) ⋯ (''x'' − ''a''''n'') + 1 has no zero in ''F''. By contrast, the fundamental theorem of algebra states that the field of complex numbers is algebraically closed. Another example of an algebraicall ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Emil Artin
Emil Artin (; March 3, 1898 – December 20, 1962) was an Austrian mathematician of Armenian descent. Artin was one of the leading mathematicians of the twentieth century. He is best known for his work on algebraic number theory, contributing largely to class field theory and a new construction of L-functions. He also contributed to the pure theories of rings, groups and fields. Along with Emmy Noether, he is considered the founder of modern abstract algebra. Early life and education Parents Emil Artin was born in Vienna to parents Emma Maria, née Laura (stage name Clarus), a soubrette on the operetta stages of Austria and Germany, and Emil Hadochadus Maria Artin, Austrian-born of mixed Austrian and Armenian descent. His Armenian last name was Artinian which was shortened to Artin. Several documents, including Emil's birth certificate, list the father's occupation as “opera singer” though others list it as “art dealer.” It seems at least plausible that he and Emma had ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Joseph Wedderburn
Joseph Henry Maclagan Wedderburn FRSE FRS (2 February 1882 – 9 October 1948) was a Scottish mathematician, who taught at Princeton University for most of his career. A significant algebraist, he proved that a finite division algebra is a field, and part of the Artin–Wedderburn theorem on simple algebras. He also worked on group theory and matrix algebra. His younger brother was the lawyer Ernest Wedderburn. Life Joseph Wedderburn was the tenth of fourteen children of Alexander Wedderburn of Pearsie, a physician, and Anne Ogilvie. He was educated at Forfar Academy then in 1895 his parents sent Joseph and his younger brother Ernest to live in Edinburgh with their paternal uncle, J R Maclagan Wedderburn, allowing them to attend George Watson's College. This house was at 3 Glencairn Crescent in the West End of the city. In 1898 Joseph entered the University of Edinburgh. In 1903, he published his first three papers, worked as an assistant in the Physical Laboratory of the U ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Simple Algebra
In abstract algebra, a branch of mathematics, a simple ring is a non-zero ring that has no two-sided ideal besides the zero ideal and itself. In particular, a commutative ring is a simple ring if and only if it is a field. The center of a simple ring is necessarily a field. It follows that a simple ring is an associative algebra over this field. So, simple algebra and ''simple ring'' are synonyms. Several references (e.g., Lang (2002) or Bourbaki (2012)) require in addition that a simple ring be left or right Artinian (or equivalently semi-simple). Under such terminology a non-zero ring with no non-trivial two-sided ideals is called quasi-simple. Rings which are simple as rings but are not a simple module over themselves do exist: a full matrix ring over a field does not have any nontrivial ideals (since any ideal of M_n(R) is of the form M_n(I) with I an ideal of R), but has nontrivial left ideals (for example, the sets of matrices which have some fixed zero columns). Accordi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Decomposition Of A Module
In abstract algebra, a decomposition of a module is a way to write a module as a direct sum of modules. A type of a decomposition is often used to define or characterize modules: for example, a semisimple module is a module that has a decomposition into simple modules. Given a ring, the types of decomposition of modules over the ring can also be used to define or characterize the ring: a ring is semisimple if and only if every module over it is a semisimple module. An indecomposable module is a module that is not a direct sum of two nonzero submodules. Azumaya's theorem states that if a module has an decomposition into modules with local endomorphism rings, then all decompositions into indecomposable modules are equivalent to each other; a special case of this, especially in group theory, is known as the Krull–Schmidt theorem. A special case of a decomposition of a module is a decomposition of a ring: for example, a ring is semisimple if and only if it is a direct sum (in fact a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Opposite Algebra
In mathematics, specifically abstract algebra, the opposite of a ring is another ring with the same elements and addition operation, but with the multiplication performed in the reverse order. More explicitly, the opposite of a ring is the ring whose multiplication ∗ is defined by for all in ''R''. The opposite ring can be used to define multimodules, a generalization of bimodules. They also help clarify the relationship between left and right modules (see '). Monoids, groups, rings, and algebras can all be viewed as categories with a single object. The construction of the opposite category generalizes the opposite group, opposite ring, etc. Relation to automorphisms and antiautomorphisms In this section the symbol for multiplication in the opposite ring is changed from asterisk to diamond, to avoid confusion with some unary operation. A ring R having isomorphic opposite ring is called a ''self-opposite'' ring, which name indicates that R^\text is essentially the same as R ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Schur's Lemma
In mathematics, Schur's lemma is an elementary but extremely useful statement in representation theory of groups and algebras. In the group case it says that if ''M'' and ''N'' are two finite-dimensional irreducible representations of a group ''G'' and ''φ'' is a linear map from ''M'' to ''N'' that commutes with the action of the group, then either ''φ'' is invertible, or ''φ'' = 0. An important special case occurs when ''M'' = ''N'', i.e. ''φ'' is a self-map; in particular, any element of the center of a group must act as a scalar operator (a scalar multiple of the identity) on ''M''. The lemma is named after Issai Schur who used it to prove the Schur orthogonality relations and develop the basics of the representation theory of finite groups. Schur's lemma admits generalisations to Lie groups and Lie algebras, the most common of which are due to Jacques Dixmier and Daniel Quillen. Representation theory of groups Representation theory is the study of homomorphi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
