# Linear topology

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, 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 if any filter basis (cf. 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

 [1] N. Bourbaki, "Elements of mathematics. Commutative algebra" , Addison-Wesley (1972) (Translated from French)