Abstract
We define topologically semiperfect (complete, separated, right linear) topological rings and characterize them by equivalent conditions. We show that the endomorphism ring of a module, endowed with the finite topology, is topologically semiperfect if and only if the module is decomposable as an (infinite) direct sum of modules with local endomorphism rings. Then we study structural properties of topologically semiperfect topological rings and prove that their topological Jacobson radicals are strongly closed and the related topological quotient rings are topologically semisimple. For the endomorphism ring of a direct sum of modules with local endomorphism rings, the topological Jacobson radical is described explicitly as the set of all matrices of nonisomorphisms. Furthermore, we prove that, over a topologically semiperfect topological ring, all finitely generated discrete modules have projective covers in the category of modules, while all lattice-finite contramodules have projective covers in both the categories of modules and contramodules. We also show that the topological Jacobson radical of a topologically semiperfect topological ring is equal to the closure of the abstract Jacobson radical, and present a counterexample demonstrating that the topological Jacobson radical can be strictly larger than the abstract one. Finally, we discuss the problem of lifting idempotents modulo the topological Jacobson radical and the structure of projective contramodules for topologically semiperfect topological rings.
Article PDF
Similar content being viewed by others
Data Availability
Data sharing not applicable to this article as no datasets were generated or analyzed during the current study.
References
Anderson, F.W,. Fuller, K.R.: Rings and categories of modules. Second edition. Graduate Texts in Mathematics 13, Springer, 92 (1974)
Angeleri Hügel, L., Saorín, M.: Modules with perfect decompositions. Math. Scand. 98, #1, p.19–43, (2006)
Bass, H.: Finitistic dimension and a homological generalization of semi-primary rings. Trans. of the Amer. Math. Soc.95, #3, p. 466–488, (1960)
Bazzoni, S., Positselski, L., Šťovíček, J.: Projective covers of flat contramodules. Internat. Math. Research Notices, 2022, #24, p. 19527–19564, (2022). 1911.11720 [math.RA]
Björk, J.E.: Rings satisfying a minimum condition on principal ideals. Journ. für die Reine und Angewandte Math.236, p. 112–119, (1969)
Camillo, V.P., Nielsen, P.P.: Half-orthogonal sets of idempotents. Trans. of the Amer. Math. Soc. 368, #2, p. 965–987, (2016)
Corner, A.L.S.: On the exchange property in additive categories. Unpublished manuscript, 60 pp. (1973)
Facchini, A.: Module theory. Endomorphism rings and direct sum decompositions in some classes of modules. Progress in Mathematics, 167/Modern Birkhäuser Classics, Birkhäuser/Springer Basel, 1998–2012
Gregorio, E.: Topologically semiperfect rings. Rendiconti Semin. Matem. Univ. Padova 85, 265–290 (1991)
Iovanov, M.C., Mesyan, Z., Reyes, M.L.: Infinite-dimensional diagonalization and semisimplicity. Israel Journ. of Math. 215, #2, p. 801–855, (2016). 1502.05184[math.RA]
Mohamed, S.H., Müller, B.: \(\aleph \)–exchange rings. “Abelian groups, module theory, and topology”, Proceedings of internat. conference in honour of A. Orsatti’s 60th birthday (Padova, 1997), Lecture Notes in Pure and Appl. Math. 201, Marcel Dekker, New York, p. 311–137 (1998)
Positselski, L.: Homological algebra of semimodules and semicontramodules: Semi-infinite homological algebra of associative algebraic structures. Appendix C in collaboration with Rumynin, D. Appendix D in collaboration with Arkhipov, S. Monografie Matematyczne vol. 70, Birkhäuser/Springer Basel, 2010. xxiv+349 pp. 0708.3398[math.CT]
Positselski, L.: Contramodules. Confluentes Math. 13, #2, p. 93–182, (2021). 1503.00991[math.CT]
Positselski, L.: Contramodules over pro-perfect topological rings. Forum Mathematicum 34, #1, p. 1–39, (2022). 1807.10671[math.CT]
Positselski, L.: Exact categories of topological vector spaces with linear topology. Electronic preprint. 2012.15431[math.CT]
Positselski, L., Příhoda, P., Trlifaj, J.: Closure properties of \(\underset{\longrightarrow }{\lim } C\). Journ. of Algebra 606, p. 30–103, (2022). 2110.13105[math.RA]
Positselski, L., Šťovíček, J.: The tilting-cotilting correspondence. Internat. Math. Research Notices 2021, #1, p. 189–274, (2021). 1710.02230[math.CT]
Positselski, L., Šťovíček, J.: Topologically semisimple and topologically perfect topological rings. Publicacions Matemàtiques 66, #2, p. 457–540, (2022). 1909.12203[math.CT]
Xu, J.: Flat covers of modules. Lecture Notes in Math. 1634, Springer, (1996)
Acknowledgements
We would like to thank the anonymous referee for careful reading of the manuscript, and particularly for suggesting the most relevant reference [9].
Funding
Open access publishing supported by the National Technical Library in Prague.
Author information
Authors and Affiliations
Corresponding author
Ethics declarations
Funding and/or conflicts of interests/competing interests
Both authors were supported by the Czech Science Foundation project 20-13778S. The first-named author is also supported by research plan RVO: 67985840 of the Czech Academy of Sciences. The authors have no competing interests to declare that are relevant to the content of this article.
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
This research was supported by GAČR project 20-13778S. The first-named author was also supported by research plan RVO: 67985840
Rights and permissions
Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.
About this article
Cite this article
Positselski, L., Šťovíček, J. Topologically Semiperfect Topological Rings. Algebr Represent Theor 27, 245–278 (2024). https://doi.org/10.1007/s10468-023-10217-x
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10468-023-10217-x