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 st… Show more
“…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.…”
Section: Introductionmentioning
confidence: 92%
“…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] …”
Section: ) the Assignement H-~h• Gives A Bijection Between The Set Omentioning
confidence: 99%
“…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,,.…”
Section: Proposition In the Notations Of 8 ~: Is The Filter Of All mentioning
“…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.…”
Section: Introductionmentioning
confidence: 92%
“…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] …”
Section: ) the Assignement H-~h• Gives A Bijection Between The Set Omentioning
confidence: 99%
“…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,,.…”
Section: Proposition In the Notations Of 8 ~: Is The Filter Of All mentioning
“…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.…”
Abstract. It is proved that if the dual Lie algebra of a Lie coalgebra is algebraic, then it is algebraic of bounded degree. This result is an analog of the D.Radford's result for associative coalgebras.
“…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.…”
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