View source for Linear topology

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''on a ring $A$''

A topology on a ring for which there is a fundamental system of neighbourhoods of zero consisting of left ideals (in this case the topology is said to be left linear). Similarly, a topology on a left $A$-module $E$ is linear if there is a fundamental system of neighbourhoods of zero consisting of submodules. The most extensively used is the [[Adic topology|adic topology]], a basis of which is given by the powers of an ideal.

A separable linearly topologized $A$-module $E$ is called a [[Linearly-compact module|linearly-compact module]] if any filter basis (cf. [[Filter|Filter]]) consisting of affine linear varieties of $E$ (that is, subsets of the form $x+E'$, where $x\in E$ and $E'$ is a submodule of $E$) has a limit point. Any module of finite type over a complete local Noetherian ring is linearly compact.

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  N. Bourbaki,   "Elements of mathematics. Commutative algebra" , Addison-Wesley  (1972)  (Translated from French)</TD></TR></table>



====Comments====
Gabriel topologies on rings are examples of linear topologies; these appear in the theory of localization (cf. [[Localization in a commutative algebra|Localization in a commutative algebra]]) or torsion theory. Gabriel topologies correspond to Serre localizing subcategories of the category of left modules over the ring.

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  J.S. Golan,   "Localization of noncommutative rings" , M. Dekker  (1975)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  B. Stenström,   "Rings of quotients" , Springer  (1975)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  F. Van Oystaeyen,   A. Verschoren,   "Reflectors and localization. Application to sheaf theory" , M. Dekker  (1979)</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  J. Lambek,   "Torsion theories, additive semantics and rings of quotients" , ''Lect. notes in math.'' , '''77''' , Springer  (1971)</TD></TR></table>