Linearly compact rings and modules

Abstract

We assume familiarity with the definitions and basic properties of linearly compact rings and modules as contained in [6] or [3, Exercises 14 20,. In § 1 we show that every linearly compact topology on a module is weaker than a unique, maximal linearly compact topology. In § 2 we apply the results of § 1 to a discussion of the circumstances under which the following statements about a linearly compact topological module E over a linearly compact ring A with radical r are equivalent:(1) the topology of E is stronger than the r-adic topology; (2) rZE is open; (3) rE is open and finitely generated; (4) r--E is open and finitely generated. Our results, applied to linearly compact rings, yield improvements of known theorems, even in the compact case. In §3 we give conditions under which a module, linearly compact for the discrete topology, is necessarily noetherian. In particular, we show that if 0 r"= (0) where r is the radical of a (not necessarily commutative) n>l ring A with identity, then A is linearly compact for the discrete topology if and only if A is (left) noetherian, A/r is (left) artinian, and A is complete for the r-adic topology. I. Maximal Linearly Compact TopologiesIf Y is a linearly compact topology on an A-module E, of all the Hausdorff linear topologies on E weaker than :-there is a weakest, which we shall denote by ,Y, [6, Theorem 2], [3, Exercises 17-18, p. 110] ; E, equipped with ~-,, is linearly compact, of course. Here we shall show that of all the linearly compact topologies on E stronger than Y, there is a strongest. We recall first that if 5: and Y are linearly compact topologies on E and if 5: is stronger than ~-, then E; 5 e and E; Y have the same closed submodules. Indeed, any subset closed for Y is clearly closed for 09~; conversely, since the identity mapping from E;5 e onto E; .Y" is continuous, a submodule closed for~ is linearly compact and hence closed for 5-. Theorem 1. Let ~--be a linearly compact topology on an A-module E.Of all the linearly compact topologies on E stronger than J there is a

A characterization of linearly compact modules

“…and a right A-module K A so that all left linearly compact R-modules are exactly those of the form Hom A (N, K), where N ranges through a convenient hereditary pretorsion class of right A-modules. Moreover, by Theorems B and C we give a new description of the finest topology p* in the equivalence class of a left linearly compact topology # on a left R-module M. This topology was already described by Warner in his paper [3] where he posed some problems concerning these arguments. As an application of our results we give a full answer to all of them.…”

“…In the example above, one should note that ~* is the finest linear topology on the left Jp-module M = J~ equivalent to # and that ~* is not the finest linear ring topology v on the ring J~ equivalent to the linear ring topology #. As a matter of fact, Warner proved in [3,Theorem 8] …”

“…Proof It is a routine work to check that ~K is the filter of all open left ideals for a left linear ring topology ~: on R (see, e.g., the proof of Theorem 7 in [3]). On the other hand it is easy to see that ~K is the finest prelinearly compact left ring topology on R equivalent to QKLet us prove that RK is an injective cogenerator for ~'Q,,.…”

Linearly compact algebraic Lie algebras and coalgebraic Lie coalgebras

“…As linearly compact spaces are Baire spaces [11] they cannot be represented by a countable number of closed subsets without interior points. This means that some A n (L) contains an interior point, say a.…”

Almost Maximal Integral Domains and Finitely Generated Modules

“…H is a ting and H =* lim R/Rr at HomR(K, K Linear compactness with respect to topologies have been studied by D. Zelinsky [20], H. Leptin [8] and [9], and S. Warner [18]. Our definition amounts to assuming that the topology on the moduleis always the discrete topology.…”