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Double Centralizer Theorem
In the branch of abstract algebra called ring theory, the double centralizer theorem can refer to any one of several similar results. These results concern the centralizer of a subring ''S'' of a ring ''R'', denoted C''R''(''S'') in this article. It is always the case that C''R''(C''R''(''S'')) contains ''S'', and a double centralizer theorem gives conditions on ''R'' and ''S'' that guarantee that C''R''(C''R''(''S'')) is ''equal'' to ''S''. Statements of the theorem Motivation The centralizer of a subring ''S'' of ''R'' is given by :\mathrm_R(S)=\.\, Clearly C''R''(C''R''(''S'')) ⊇ ''S'', but it is not always the case that one can say the two sets are equal. The double centralizer theorems give conditions under which one can conclude that equality occurs. There is another special case of interest. Let ''M'' be a right ''R'' module and give ''M'' the natural left ''E''-module structure, where ''E'' is End(''M''), the ring of endomorphisms of the abelian group ''M''. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Abstract Algebra
In mathematics, more specifically algebra, abstract algebra or modern algebra is the study of algebraic structures. Algebraic structures include groups, rings, fields, modules, vector spaces, lattices, and algebras over a field. The term ''abstract algebra'' was coined in the early 20th century to distinguish this area of study from older parts of algebra, and more specifically from elementary algebra, the use of variables to represent numbers in computation and reasoning. Algebraic structures, with their associated homomorphisms, form mathematical categories. Category theory is a formalism that allows a unified way for expressing properties and constructions that are similar for various structures. Universal algebra is a related subject that studies types of algebraic structures as single objects. For example, the structure of groups is a single object in universal algebra, which is called the ''variety of groups''. History Before the nineteenth century, algebra meant ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Primitive Ring
In the branch of abstract algebra 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 spaces and Weyl algebras over fields of characteristic zero. Definition A ring ''R'' is said to be a left primitive ring if it has a faithful simple left ''R''-module. A right primitive ring is defined similarly with right ''R''-modules. There are rings which are primitive on one side but not on the other. The first example was constructed by George M. Bergman in . Another example found by Jategaonkar showing the distinction can be found in . An internal characterization of left primitive rings is as follows: a ring is left primitive if and only if there is a maximal left ideal containing no nonzero two-sided ideals. The analogous definition for right primitive rings is also valid. The structure of left primitive rings is completely determined by the Jacobson density theorem: A ring is l ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Balanced Module
In the subfield of abstract algebra known as module theory, a right ''R'' module ''M'' is called a balanced module (or is said to have the double centralizer property) if every endomorphism of the abelian group ''M'' which commutes with all ''R''-endomorphisms of ''M'' is given by multiplication by a ring element. Explicitly, for any additive endomorphism ''f'', if ''fg'' = ''gf'' for every ''R'' endomorphism ''g'', then there exists an ''r'' in ''R'' such that ''f''(''x'') = ''xr'' for all ''x'' in ''M''. In the case of non-balanced modules, there will be such an ''f'' that is not expressible this way. In the language of centralizers, a balanced module is one satisfying the conclusion of the double centralizer theorem, that is, the only endomorphisms of the group ''M'' commuting with all the ''R'' endomorphisms of ''M'' are the ones induced by right multiplication by ring elements. A ring is called balanced if every right ''R'' module is balanced.The definitions ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Weak Operator Topology
In functional analysis, the weak operator topology, often abbreviated WOT, is the weakest topology on the set of bounded operators on a Hilbert space H, such that the functional sending an operator T to the complex number \langle Tx, y\rangle is continuous for any vectors x and y in the Hilbert space. Explicitly, for an operator T there is base of neighborhoods of the following type: choose a finite number of vectors x_i, continuous functionals y_i, and positive real constants \varepsilon_i indexed by the same finite set I. An operator S lies in the neighborhood if and only if , y_i(T(x_i) - S(x_i)), 0. Relationships between different topologies on ''B(X,Y)'' The different terminology for the various topologies on B(X,Y) can sometimes be confusing. For instance, "strong convergence" for vectors in a normed space sometimes refers to norm-convergence, which is very often distinct from (and stronger than) than SOT-convergence when the normed space in question is B(X,Y). The w ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Von Neumann Algebra
In mathematics, a von Neumann algebra or W*-algebra is a *-algebra of bounded operators on a Hilbert space that is closed in the weak operator topology and contains the identity operator. It is a special type of C*-algebra. Von Neumann algebras were originally introduced by John von Neumann, motivated by his study of single operators, group representations, ergodic theory and quantum mechanics. His double commutant theorem shows that the analytic definition is equivalent to a purely algebraic definition as an algebra of symmetries. Two basic examples of von Neumann algebras are as follows: *The ring L^\infty(\mathbb R) of essentially bounded measurable functions on the real line is a commutative von Neumann algebra, whose elements act as multiplication operators by pointwise multiplication on the Hilbert space L^2(\mathbb R) of square-integrable functions. *The algebra \mathcal B(\mathcal H) of all bounded operators on a Hilbert space \mathcal H is a von Neumann algebr ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hilbert Space
In mathematics, Hilbert spaces (named after David Hilbert) allow generalizing the methods of linear algebra and calculus from (finite-dimensional) Euclidean vector spaces to spaces that may be infinite-dimensional. Hilbert spaces arise naturally and frequently in mathematics and physics, typically as function spaces. Formally, a Hilbert space is a vector space equipped with an inner product that defines a distance function for which the space is a complete metric space. The earliest Hilbert spaces were studied from this point of view in the first decade of the 20th century by David Hilbert, Erhard Schmidt, and Frigyes Riesz. They are indispensable tools in the theories of partial differential equations, quantum mechanics, Fourier analysis (which includes applications to signal processing and heat transfer), and ergodic theory (which forms the mathematical underpinning of thermodynamics). John von Neumann coined the term ''Hilbert space'' for the abstract concept that under ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Bounded Operators
In functional analysis and operator theory, a bounded linear operator is a linear transformation L : X \to Y between topological vector spaces (TVSs) X and Y that maps bounded subsets of X to bounded subsets of Y. If X and Y are normed vector spaces (a special type of TVS), then L is bounded if and only if there exists some M > 0 such that for all x \in X, \, Lx\, _Y \leq M \, x\, _X. The smallest such M is called the operator norm of L and denoted by \, L\, . A bounded operator between normed spaces is continuous and vice versa. The concept of a bounded linear operator has been extended from normed spaces to all topological vector spaces. Outside of functional analysis, when a function f : X \to Y is called " bounded" then this usually means that its image f(X) is a bounded subset of its codomain. A linear map has this property if and only if it is identically 0. Consequently, in functional analysis, when a linear operator is called "bounded" then it is never meant in this a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Von Neumann Bicommutant Theorem
In mathematics, specifically functional analysis, the von Neumann bicommutant theorem relates the closure of a set of bounded operators on a Hilbert space in certain topologies to the bicommutant of that set. In essence, it is a connection between the algebraic and topological sides of operator theory. The formal statement of the theorem is as follows: :Von Neumann bicommutant theorem. Let be an algebra consisting of bounded operators on a Hilbert space , containing the identity operator, and closed under taking adjoints. Then the closures of in the weak operator topology and the strong operator topology are equal, and are in turn equal to the bicommutant of . This algebra is called the von Neumann algebra generated by . There are several other topologies on the space of bounded operators, and one can ask what are the *-algebras closed in these topologies. If is closed in the norm topology then it is a C*-algebra, but not necessarily a von Neumann algebra. One such ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Simple Ring
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). Accord ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Center Of A Ring
In algebra, the center of a ring ''R'' is the subring consisting of the elements ''x'' such that ''xy = yx'' for all elements ''y'' in ''R''. It is a commutative ring and is denoted as Z(R); "Z" stands for the German word ''Zentrum'', meaning "center". If ''R'' is a ring, then ''R'' is an associative algebra over its center. Conversely, if ''R'' is an associative algebra over a commutative subring ''S'', then ''S'' is a subring of the center of ''R'', and if ''S'' happens to be the center of ''R'', then the algebra ''R'' is called a central algebra. Examples *The center of a commutative ring ''R'' is ''R'' itself. *The center of a skew-field is a field. *The center of the (full) matrix ring with entries in a commutative ring ''R'' consists of ''R''-scalar multiples of the identity matrix. *Let ''F'' be a field extension of a field ''k'', and ''R'' an algebra over ''k''. Then Z\left(R \otimes_k F\right) = Z(R) \otimes_k F. *The center of the universal enveloping algebra of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Polynomial Identity Ring
In ring theory, a branch of mathematics, a ring ''R'' is a polynomial identity ring if there is, for some ''N'' > 0, an element ''P'' ≠ 0 of the free algebra, Z, over the ring of integers in ''N'' variables ''X''1, ''X''2, ..., ''X''''N'' such that :P(r_1, r_2, \ldots, r_N) = 0 for all ''N''-tuples ''r''1, ''r''2, ..., ''r''''N'' taken from ''R''. Strictly the ''X''''i'' here are "non-commuting indeterminates", and so "polynomial identity" is a slight abuse of language, since "polynomial" here stands for what is usually called a "non-commutative polynomial". The abbreviation PI-ring is common. More generally, the free algebra over any ring ''S'' may be used, and gives the concept of PI-algebra. If the degree of the polynomial ''P'' is defined in the usual way, the polynomial ''P'' is called monic if at least one of its terms of highest degree has coefficient equal to 1. Every commutative ring is a PI-ring, satisfying the polynomial identity ''XY'' − ''YX'' = 0. Therefore, PI- ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Jacobson Density Theorem
In mathematics, more specifically non-commutative ring theory, modern algebra, and module theory, the Jacobson density theorem is a theorem concerning simple modules over a ring . The theorem can be applied to show that any primitive ring can be viewed as a "dense" subring of the ring of linear transformations of a vector space.Isaacs, Corollary 13.16, p. 187 This theorem first appeared in the literature in 1945, in the famous paper "Structure Theory of Simple Rings Without Finiteness Assumptions" by Nathan Jacobson. This can be viewed as a kind of generalization of the Artin-Wedderburn theorem's conclusion about the structure of simple Artinian rings. Motivation and formal statement Let be a ring and let be a simple right -module. If is a non-zero element of , (where is the cyclic submodule of generated by ). Therefore, if are non-zero elements of , there is an element of that induces an endomorphism of transforming to . The natural question now is whether this can be ge ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
