Linearly-compact module
A topological module over a topological ring that has a basis (cf. Base) of neighbourhoods of zero consisting of submodules, and in which every centred system (or filter base, cf. also 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). 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. 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
| [1] | S. Lefschetz, "Algebraic topology" , Amer. Math. Soc. (1955) |
| [2] | D. Zelinsky, "Linearly compact modules and rings" Amer. J. Math. , 75 : 1 (1953) pp. 79–90 |
| [3a] | H. Leptin, "Linear kompakte Moduln und Ringe" Math. Z. , 62 (1955) pp. 241–267 |
| [3b] | H. Leptin, "Linear kompakte Moduln und Ringe II" Math. Z. , 66 (1957) pp. 289–327 |
Comments
A topological module 
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 
of 
is linearly compact if for every system 
of closed linear subvarieties of 
with the finite intersection property, i.e. every finite intersection of elements of 
is non-empty, the intersection 
is non-empty. Such a linear subvariety is closed.
A filtration on a group 
(indexed by 
) is an increasing or decreasing collection of subgroups 
of 
. Let 
be a ring with a filtration 
of the underlying additive group of 
. Such a filtration is said to be compatible with the ring structure if 
for all 
and 
. A filtered ring is a ring provided with such a filtration. Now let 
be an 
-module and let 
be a filtration of the underlying Abelian group. This filtration is said to be compatible with the filtered ring 
if 
for all 
. Such a module is called a filtered module over the filtered ring 
. If 
, as is often the case, the 
are all 
-submodules of 
. The definitions in the article filtered module correspond to the case that all 
are indeed 
-submodules of 
. Using the 
and 
as a base of open neighbourhoods in 
and 
, linear topologies are defined on 
and 
. These are the more frequently occurring examples of linearly topologized modules and rings.
References
| [a1] | N. Bourbaki, "Algèbre commutative" , Eléments de mathématiques , Hermann (1961) pp. Chapt. 3. Graduations, filtrations, et topologies |
| [a2] | S. Warner, "Topological fields" , North-Holland (1989) pp. Chapt. 5, Sect.31 |
Linearly-compact module. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Linearly-compact_module&oldid=43463