|
Depth Two Subring
In ring theory and Frobenius algebra extensions, areas of mathematics, there is a notion of depth two subring or depth of a Frobenius extension. The notion of depth two is important in a certain noncommutative Galois theory, which generates Hopf algebroids in place of the more classical Galois groups, whereas the notion of depth greater than two measures 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 subring in any associative ring is proposed (see below) in a paper studying the depth of a subgroup of a finite group as group algebras over a commutative ring. Definition and first examples A unital subring B \subseteq A has (or is) right depth two if there is a split epimorphism of natural ''A''-''B''-bimodules from A^n \rightarrow A \otimes_B A for some positive integer ''n''; by switching to natural ''B''-''A''-bimodules, there is a corresponding definition of left d ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hopf Algebra
In mathematics, a Hopf algebra, named after Heinz Hopf, is a structure that is simultaneously a ( unital associative) algebra and a (counital coassociative) coalgebra, with these structures' compatibility making it a bialgebra, and that moreover is equipped with an antihomomorphism satisfying a certain property. The representation theory of a Hopf algebra is particularly nice, since the existence of compatible comultiplication, counit, and antipode allows for the construction of tensor products of representations, trivial representations, and dual representations. Hopf algebras occur naturally in algebraic topology, where they originated and are related to the H-space concept, in group scheme theory, in group theory (via the concept of a group ring), and in numerous other places, making them probably the most familiar type of bialgebra. Hopf algebras are also studied in their own right, with much work on specific classes of examples on the one hand and classification problems o ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Homological Algebra
Homological algebra is the branch of mathematics that studies homology (mathematics), homology in a general algebraic setting. It is a relatively young discipline, whose origins can be traced to investigations in combinatorial topology (a precursor to algebraic topology) and abstract algebra (theory of module (mathematics), modules and Syzygy (mathematics), syzygies) at the end of the 19th century, chiefly by Henri Poincaré and David Hilbert. Homological algebra is the study of homological functors and the intricate algebraic structures that they entail; its development was closely intertwined with the emergence of category theory. A central concept is that of chain complexes, which can be studied through their homology and cohomology. Homological algebra affords the means to extract information contained in these complexes and present it in the form of homological invariant (mathematics), invariants of ring (mathematics), rings, modules, topological spaces, and other "tangible ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Galois Theory
In mathematics, Galois theory, originally introduced by Évariste Galois, provides a connection between field (mathematics), field theory and group theory. This connection, the fundamental theorem of Galois theory, allows reducing certain problems in field theory to group theory, which makes them simpler and easier to understand. Galois introduced the subject for studying root of a function, roots of polynomials. This allowed him to characterize the polynomial equations that are solvable by radicals in terms of properties of the permutation group of their roots—an equation is by definition ''solvable by radicals'' if its roots may be expressed by a formula involving only integers, nth root, th roots, and the four basic arithmetic operations. This widely generalizes the Abel–Ruffini theorem, which asserts that a general polynomial of degree at least five cannot be solved by radicals. Galois theory has been used to solve classic problems including showing that two problems of a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Bratteli Diagram
In mathematics, a Bratteli diagram is a combinatorial structure: a Graph (discrete mathematics), graph composed of vertices labelled by positive integers ("level") and unoriented edges between vertices having levels differing by one. The notion was introduced by Ola Bratteli in 1972 in the theory of operator algebras to describe directed sequences of finite-dimensional algebras: it played an important role in Elliott's classification of approximately finite-dimensional C*-algebra, AF-algebras and the theory of subfactors. Subsequently Anatoly Vershik associated dynamical systems with infinite paths in such graphs. Definition A Bratteli diagram is given by the following objects: * A sequence of sets ''V''''n'' ('the vertices at level ''n'' ') labeled by positive integer set N. In some literature each element v of ''V''''n'' is accompanied by a positive integer ''b''''v'' > 0. * A sequence of sets ''E''''n'' ('the edges from level ''n'' to ''n'' + 1 ') ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Annihilator (ring Theory)
In mathematics, the annihilator of a subset of a module over a ring is the ideal formed by the elements of the ring that give always zero when multiplied by each element of . Over an integral domain, a module that has a nonzero annihilator is a torsion module, and a finitely generated torsion module has a nonzero annihilator. The above definition applies also in the case of noncommutative rings, where the left annihilator of a left module is a left ideal, and the right-annihilator, of a right module is a right ideal. Definitions Let ''R'' be a ring, and let ''M'' be a left ''R''- module. Choose a non-empty subset ''S'' of ''M''. The ''annihilator'' of ''S'', denoted Ann''R''(''S''), is the set of all elements ''r'' in ''R'' such that, for all ''s'' in ''S'', . In set notation, :\mathrm_R(S)=\ It is the set of all elements of ''R'' that "annihilate" ''S'' (the elements for which ''S'' is a torsion set). Subsets of right modules may be used as well, after the modifi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Semisimple Algebra
In ring theory, a branch of mathematics, a semisimple algebra is an associative Artinian algebra over a field which has trivial Jacobson radical (only the zero element of the algebra is in the Jacobson radical). If the algebra is finite-dimensional this is equivalent to saying that it can be expressed as a Cartesian product of simple subalgebras. Definition The Jacobson radical of an algebra over a field is the ideal consisting of all elements that annihilate every simple left-module. The radical contains all nilpotent ideals, and if the algebra is finite-dimensional, the radical itself is a nilpotent ideal. A finite-dimensional algebra is then said to be ''semisimple'' if its radical contains only the zero element. An algebra ''A'' is called ''simple'' if it has no proper ideals and ''A''2 = ≠ . As the terminology suggests, simple algebras are semisimple. The only possible ideals of a simple algebra ''A'' are ''A'' and . Thus if ''A'' is simple, then ''A'' is not nilpote ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
G-set
In mathematics, a group action of a group G on a set S is a group homomorphism from G to some group (under function composition) of functions from S to itself. It is said that G acts on S. Many sets of transformations form a group under function composition; for example, the rotations around a point in the plane. It is often useful to consider the group as an abstract group, and to say that one has a group action of the abstract group that consists of performing the transformations of the group of transformations. The reason for distinguishing the group from the transformations is that, generally, a group of transformations of a structure acts also on various related structures; for example, the above rotation group also acts on triangles by transforming triangles into triangles. If a group acts on a structure, it will usually also act on objects built from that structure. For example, the group of Euclidean isometries acts on Euclidean space and also on the figures drawn i ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Generator (category Theory)
In mathematics, specifically category theory, a family of generators (or family of separators) of a category \mathcal C is a collection \mathcal G \subseteq Ob(\mathcal C) of objects in \mathcal C, such that for any two ''distinct'' morphisms f, g: X \to Y in \mathcal, that is with f \neq g, there is some G in \mathcal G and some morphism h : G \to X such that f \circ h \neq g \circ h. If the collection consists of a single object G, we say it is a generator (or separator). Generators are central to the definition of Grothendieck categories. The dual concept is called a cogenerator (or coseparator). Examples * In the category of abelian groups, the group of integers \mathbb Z is a generator: If ''f'' and ''g'' are different, then there is an element x \in X, such that f(x) \neq g(x). Hence the map \mathbb Z \rightarrow X, n \mapsto n \cdot x suffices. * Similarly, the one-point set is a generator for the category of sets. In fact, any nonempty set is a generator. * In the categ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Projective Module
In mathematics, particularly in algebra, the class of projective modules enlarges the class of free modules (that is, modules with basis vectors) over a ring, keeping some of the main properties of free modules. Various equivalent characterizations of these modules appear below. Every free module is a projective module, but the converse fails to hold over some rings, such as Dedekind rings that are not principal ideal domains. However, every projective module is a free module if the ring is a principal ideal domain such as the integers, or a (multivariate) polynomial ring over a field (this is the Quillen–Suslin theorem). Projective modules were first introduced in 1956 in the influential book ''Homological Algebra'' by Henri Cartan and Samuel Eilenberg. Definitions Lifting property The usual category theoretical definition is in terms of the property of ''lifting'' that carries over from free to projective modules: a module ''P'' is projective if and only if fo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
