mathematics, a module over a (not necessarily commutative) ring is countably generated if it is generated as a module by a countable subset. The importance...

generating set Countably generated space, a topological space in which the topology is determined by its countable subsets Countably generated module. (Kaplansky's...

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...

a direct product of countably many copies of F2 and I is the direct sum of countably many copies of F2 inside of R. The R-module R/I is locally free since...

a module. Countably generated module Flat module Invariant basis number "ac.commutative algebra – Existence of a minimal generating set of a module –...

projective module over an arbitrary ring is a direct sum of countably generated projective modules. Show that a countably generated projective module over a...

convergent sequences. The countably generated spaces are precisely the spaces having countable tightness—therefore the name countably tight is used as well...

_{C(X)}(x):=g(\sigma (x),\rho (x)).} The converse holds as well: Every countably generated Hilbert C*-module over a commutative unital C*-algebra A = C ( X ) {\displaystyle...

cardinality of a generating set for the group. By definition, the rank of a finitely generated group is finite. Finitely generated module Presentation of...

of cyclic groups. Every finite abelian group is finitely generated. The finitely generated abelian groups can be completely classified. The integers...

modules is torsionless. A free module is reflexive if it is finitely generated, and for some rings there are also infinitely generated free modules that...

continuous module countably generated A countably generated module is a module that admits a generating set whose cardinality is at most countable. cyclic...

\dots ,x_{n}]} is finitely generated. The polynomial algebra in countably infinitely many generators is infinitely generated. The field E = K ( t ) {\displaystyle...

topology – Type of topology Countably generated space – topological space in which the topology is determined by its countable subsetsPages displaying wikidata...

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...

σ-locally finite base. A σ-locally finite base is a base which is a union of countably many locally finite collections of open sets. For a closely related theorem...

every sequence has a convergent subsequence. Countably compact. A space is countably compact if every countable open cover has a finite subcover. Pseudocompact...

Von Neumann algebras are semihereditary: every finitely generated submodule of a projective module is itself projective. There have been several attempts...

(or, equivalently, from closed sets) through the operations of countable union, countable intersection, and relative complement. Borel sets are named after...

of a finite number of Fréchet spaces is a Fréchet space. A product of countably many Fréchet spaces is always once again a Fréchet space. However, an...

separably generated if it admits a separating transcendence basis. If a field extension is finitely generated and it is also separably generated, then each...

space X {\displaystyle X} is said to be limit point compact or weakly countably compact if every infinite subset of X {\displaystyle X} has a limit point...

classification of finitely generated abelian groups which is a specialization of the structure theorem for finitely generated modules over a principal ideal...

Locally compact space Compactly generated space Axiom of countability Sequential space First-countable space Second-countable space Separable space Lindelöf...

hereditary, so any submodule generated by injective modules is injective. The converse is a result of (Matlis 1958): if every module has a unique maximal injective...

this we can deduce that there are (up to isomorphism) only countably many finitely generated recursively presented groups. Bernhard Neumann has shown that...

retained. At the next step a new vector is generated in the same hypercube, and its angles with the previously generated vectors are evaluated. If these angles...

vector lattice. An ordered vector space is said to be countably order complete if each countable subset that is bounded above has a supremum. Being an...

the Jacobson radical. Every finitely generated algebra over R that is a field is finitely generated as an R-module. (Zariski's lemma) Every prime ideal...

to show that every finitely generated abelian group is finitely presented. For, if G {\displaystyle G} is finitely generated by a set B {\displaystyle B}...