El Bashir, Robert; Kepka, Tomáš Notes on slender prime rings. (English) Zbl 0854.16016 Commentat. Math. Univ. Carol. 37, No. 2, 419-422 (1996). For a (non-commutative) prime ring \(R\) the following hold: (1) If every right ideal is an ideal and \(R\) (as a right \(R\)-module) does not have any non-discrete metrizable linear topology, then \(R\) is a domain and \(R\) is slender (as a left \(R\)-module), iff \(R\) is not complete in a nondiscrete metrizable filtration, as a left \(R\)-module. (2) If the additive group \((R,+)\) is not complete in a non-discrete metrizable linear topology, then \(R\) is slender iff \(R\) is not isomorphic to a full matrix ring over a division ring. (3) If cardinality of \(R\) is \(\geq 2^{\aleph_0}\), and the additive group \((R,+)\) is not complete in non-discrete metrizable topology, then \(R\) is slender.Reviewer’s remark: Cf. the reviewer’s following result from 1982 [Abelian Group Theory, Lect. Notes Math. 1006, 375-383 (1983; Zbl 0517.18013)]: For any ring \(R\) and any \(R\)-module \(M\), \(M\) is slender if and only if: (1) \(\text{Hom}_R(\prod Ra_n/\coprod Ra_n,M)=0\), for every countable family \(Ra_n\) of cyclic modules, (2) \(\prod Ra_n\) is not a submodule of \(M\), for any countable set of cyclic \(Ra_n\) and (3) \(M\) does not contain a submodule which is a completion of a cyclic module in any non-discrete metrizable linear topology. Reviewer: R.Dimitrić (Berkeley) MSC: 16N60 Prime and semiprime associative rings 16D70 Structure and classification for modules, bimodules and ideals (except as in 16Gxx), direct sum decomposition and cancellation in associative algebras) 16W60 Valuations, completions, formal power series and related constructions (associative rings and algebras) 16W80 Topological and ordered rings and modules Keywords:slender rings; prime rings; linear topologies; metrizable filtrations; additive groups; cyclic modules Citations:Zbl 0517.18013 × Cite Format Result Cite Review PDF Full Text: EuDML