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Linearly Topologized Ring
In algebra, a linear topology on a left A-module M is a topology on M that is invariant under translations and admits a fundamental system of neighborhood of 0 that consists of submodules of M. If there is such a topology, M is said to be linearly topologized. If A is given a discrete topology, then M becomes a topological A-module with respect to a linear topology. See also * * * * * * * * * References * Bourbaki, N. (1972). Commutative algebra (Vol. 8). Hermann. Topology Topological algebra Topological groups {{algebra-stub ... [...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] |
Topology
In mathematics, topology (from the Greek language, Greek words , and ) is concerned with the properties of a mathematical object, geometric object that are preserved under Continuous function, continuous Deformation theory, deformations, such as Stretch factor, stretching, Twist (mathematics), twisting, crumpling, and bending; that is, without closing holes, opening holes, tearing, gluing, or passing through itself. A topological space is a set (mathematics), set endowed with a structure, called a ''Topology (structure), topology'', which allows defining continuous deformation of subspaces, and, more generally, all kinds of continuity (mathematics), continuity. Euclidean spaces, and, more generally, metric spaces are examples of a topological space, as any distance or metric defines a topology. The deformations that are considered in topology are homeomorphisms and homotopy, homotopies. A property that is invariant under such deformations is a topological property. Basic exampl ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Topological Module
In mathematics, a topological module is a module over a topological ring such that scalar multiplication and addition are continuous. Examples A topological vector space is a topological module over a topological field. An abelian topological group can be considered as a topological module over \Z, where \Z is the ring of integers with the discrete topology. A topological ring is a topological module over each of its subring In mathematics, a subring of ''R'' is a subset of a ring that is itself a ring when binary operations of addition and multiplication on ''R'' are restricted to the subset, and which shares the same multiplicative identity as ''R''. For those wh ...s. A more complicated example is the I-adic topology on a ring and its modules. Let I be an ideal of a ring R. The sets of the form x + I^n for all x \in R and all positive integers n, form a base for a topology on R that makes R into a topological ring. Then for any left R-module M, the sets of the f ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Topology
In mathematics, topology (from the Greek language, Greek words , and ) is concerned with the properties of a mathematical object, geometric object that are preserved under Continuous function, continuous Deformation theory, deformations, such as Stretch factor, stretching, Twist (mathematics), twisting, crumpling, and bending; that is, without closing holes, opening holes, tearing, gluing, or passing through itself. A topological space is a set (mathematics), set endowed with a structure, called a ''Topology (structure), topology'', which allows defining continuous deformation of subspaces, and, more generally, all kinds of continuity (mathematics), continuity. Euclidean spaces, and, more generally, metric spaces are examples of a topological space, as any distance or metric defines a topology. The deformations that are considered in topology are homeomorphisms and homotopy, homotopies. A property that is invariant under such deformations is a topological property. Basic exampl ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Topological Algebra
In mathematics, a topological algebra A is an algebra and at the same time a topological space, where the algebraic and the topological structures are coherent in a specified sense. Definition A topological algebra A over a topological field K is a topological vector space together with a bilinear multiplication :\cdot: A \times A \to A, :(a,b) \mapsto a \cdot b that turns A into an algebra over K and is continuous in some definite sense. Usually the ''continuity of the multiplication'' is expressed by one of the following (non-equivalent) requirements: * ''joint continuity'': for each neighbourhood of zero U\subseteq A there are neighbourhoods of zero V\subseteq A and W\subseteq A such that V \cdot W\subseteq U (in other words, this condition means that the multiplication is continuous as a map between topological spaces or * ''stereotype continuity'': for each totally bounded set S\subseteq A and for each neighbourhood of zero U\subseteq A there is a neighbourhood of zero V\su ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
