View source for Linearly-compact module

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A [[Topological module|topological module]] over a [[Topological ring|topological ring]] that has a basis (cf. [[Base|Base]]) of neighbourhoods of zero consisting of submodules, and in which every centred system (or filter base, cf. also [[Centred family of sets|Centred family of sets]]) consisting of cosets with respect to closed submodules has a non-empty intersection. Every linearly-compact module is a complete topological group.

A linearly-compact module is called a linearly-compact module in the narrow sense if every continuous homomorphism onto a topological module that has a basis of neighbourhoods of zero consisting of submodules, is open (cf. [[Open mapping|Open mapping]]). A topological module is a linearly-compact module in the narrow sense if and only if it is a complete topological group and if every quotient module of it with respect to an open submodule is an [[Artinian module|Artinian module]]. In particular, an Artinian module in the discrete topology is a linearly-compact module in the narrow sense. Thus, linearly-compact modules in the narrow sense are the topological analogues of Artinian modules.

Direct products, closed submodules, quotient modules with respect to closed submodules, and continuous homomorphic images having a basis of neighbourhoods of zero consisting of submodules of linearly-compact modules (in the narrow sense) are themselves linearly-compact modules (in the narrow sense).

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  S. Lefschetz,   "Algebraic topology" , Amer. Math. Soc.  (1955)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  D. Zelinsky,   "Linearly compact modules and rings"  ''Amer. J. Math.'' , '''75''' :  1  (1953)  pp. 79–90</TD></TR><TR><TD valign="top">[3a]</TD> <TD valign="top">  H. Leptin,   "Linear kompakte Moduln und Ringe"  ''Math. Z.'' , '''62'''  (1955)  pp. 241–267</TD></TR><TR><TD valign="top">[3b]</TD> <TD valign="top">  H. Leptin,   "Linear kompakte Moduln und Ringe II"  ''Math. Z.'' , '''66'''  (1957)  pp. 289–327</TD></TR></table>



====Comments====
A topological module $M$ over a (topological) field (ring) is said to have a linear topology if there is a base of neighbourhoods of zero consisting of submodules. A linear subvariety $V$ of $M$ is linearly compact if for every system $\{V_\alpha\}$ of closed linear subvarieties of $V$ with the finite intersection property, i.e. every finite intersection of elements of $\{V_\alpha\}$ is non-empty, the intersection $\bigcap V_\alpha$ is non-empty. Such a linear subvariety is closed.

A filtration on a group $G$ (indexed by $\mathbf Z$) is an increasing or decreasing collection of subgroups $(G_n)_{n\in\mathbf Z}$ of $G$. Let $A$ be a ring with a filtration $(A_n)_{n\in\mathbf Z}$ of the underlying additive group of $A$. Such a filtration is said to be compatible with the ring structure if $A_nA_m\subset A_{n+m}$ for all $n,m\in\mathbf Z$ and $1\in A_0$. A filtered ring is a ring provided with such a filtration. Now let $M$ be an $A$-module and let $(M_n)_{n\in\mathbf Z}$ be a filtration of the underlying Abelian group. This filtration is said to be compatible with the filtered ring $A$ if $A_nM_m\subset M_{n+m}$ for all $n,m\in\mathbf Z$. Such a module is called a filtered module over the filtered ring $A$. If $A_0=A$, as is often the case, the $M_n$ are all $A$-submodules of $M$. The definitions in the article [[Filtered module|filtered module]] correspond to the case that all $M_n$ are indeed $A$-submodules of $M$. Using the $(A_n)$ and $(M_n)$ as a base of open neighbourhoods in $A$ and $M$, linear topologies are defined on $A$ and $M$. These are the more frequently occurring examples of linearly topologized modules and rings.

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  N. Bourbaki,   "Algèbre commutative" , ''Eléments de mathématiques'' , Hermann  (1961)  pp. Chapt. 3. Graduations, filtrations, et topologies</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  S. Warner,   "Topological fields" , North-Holland  (1989)  pp. Chapt. 5, Sect.31</TD></TR></table>