Abstract
We study some closely interrelated notions of Homological Algebra: (1) We define a topology on modules over a not-necessarily commutative ring R that coincides with the R-topology defined by Matlis when R is commutative. (2) We consider the class \( \mathcal {SF}\) of strongly flat modules when R is a right Ore domain with classical right quotient ring Q. Strongly flat modules are flat. The completion of R in its R-topology is a strongly flat R-module. (3) We prove some results related to the question whether \( \mathcal {SF}\) a covering class implies \( \mathcal {SF}\) closed under direct limits. This is a particular case of the so-called Enochs’ Conjecture (whether covering classes are closed under direct limits). Some of our results concern right chain domains. For instance, we show that if the class of strongly flat modules over a right chain domain R is covering, then R is right invariant. In this case, flat R-modules are strongly flat.
Similar content being viewed by others
References
Amini, B., Amini, A., Facchini, A.: Equivalence of diagonal matrices over local rings. J. Algebra 320, 1288–1310 (2008)
Anderson, D.W., Fuller, K.R.: Rings and categories of modules, 2nd edn. Springer, New York (1992)
Angeleri Hügel, L., Sánchez, J.: Tilting modules arising from ring epimorphisms. Algebr. Represent. Theor. 14, 217–246 (2011)
Bazzoni, S., Salce, L.: Strongly flat covers. J. Lond. Math. Soc. 66, 276–294 (2002)
Bazzoni, S., Positselski, L.: \(S\)-almost perfect commutative rings. J. Algebra 532, 323–356 (2019)
Bazzoni, S., Positselski, L.: Contramodules over pro-perfect topological rings, the covering property in categorical tilting theory, and homological ring epimorphisms, available in arXiv:1807.10671
Bessenrodt, C., Brungs, H. H., Törner, G.: Right chain rings, Part 1, Schriftenreihe des Fachbereichs Math. 181 (Universität Duisburg, 1990)
Brungs, H.H., Dubrovin, N.I.: A classification and examples of rank one chain domains. Trans. Am. Math. Soc. 355, 2733–2753 (2003)
Cohn, P.: Free ideal rings and localizations in general rings. Cambridge University Press, Cambridge (2006)
Dung, N.V., Facchini, A.: Direct summands of serial modules. J. Pure Appl. Algebra 133, 93–106 (1998)
Dung, N.V., Facchini, A.: Weak Krull–Schmidt for infinite direct sums of uniserial modules. J. Algebra 193, 102–121 (1997)
Facchini, A.: Krull–Schmidt fails for serial modules. Trans. Am. Math. Soc. 348, 4561–4576 (1996)
Facchini, A., Nazemian, Z.: Equivalence of some homological conditions for ring epimorphism. J. Pure Appl. Algebra 223, 1440–1455 (2019)
Facchini, A., Salce, L.: Uniserial modules: sums and isomorphisms of subquotients. Comm. Algebra 18(2), 499–517 (1990)
Fuchs, L., Salce, L.: Almost perfect commutative rings. J. Pure Appl. Algebra 222, 4223–4238 (2018)
Goodearl, K.R.: Ring theory; nonsingular rings and modules. Dekker, New York (1976)
Goodearl, K.R., Warfield, R.B.: An introduction to noncommutative noetherian rings, 2nd edn. Cambridge Univ. Press, Cambridge (2004)
Göbel, R., Trlifaj, J.: Approximations and endomorphism algebras of modules. Walter de Gruyter, Berlin (2006)
Lam, T.Y.: Lectures on modules and rings. Springer, New York (1999)
Matlis, E.: 1 -dimensional Cohen–Macaulay rings. Springer, Berlin, New York (1973)
Nicholson, W.K., Yousif, M.F.: Quasi-Frobenius rings. Cambridge University Press, Cambridge (2003)
Positselski, L.: Flat ring epimorphisms of countable type, available in arXiv:1808.00937
Příhoda, P.: \({\rm Add}(U)\) of a uniserial module. Comment. Math. Univ. Carolin. 47, 391–398 (2006)
Puninski, G.: Some model theory over a nearly simple uniserial domain and decompositions of serial modules. J. Pure Appl. Algebra 163, 319–337 (2001)
Stenström, B.: Rings of quotients. Springer, New York (1975)
Schofield, A.H.: Representations of rings over skew fields. Cambridge University Press, Cambridge (1985)
Warfield, R.B.: Purity and algebraic compactness for modules. Pacific J. Math. 28, 699–719 (1969)
Wisbauer, R.: Foundations of module and ring theory. Gordon and Breach, Philadelphia (1991)
Xu, J.: Flat covers of modules. Lecture notes in mathematics, vol. 1634. Springer, New York (1996)
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
The first author was partially supported by Dipartimento di Matematica “Tullio Levi-Civita” of Università di Padova (Project BIRD163492/16 “Categorical homological methods in the study of algebraic structures” and Research program DOR1714214 “Anelli e categorie di moduli”). The second author was supported by a Grant from IPM.
Rights and permissions
About this article
Cite this article
Facchini, A., Nazemian, Z. Covering classes, strongly flat modules, and completions. Math. Z. 296, 239–259 (2020). https://doi.org/10.1007/s00209-019-02417-3
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00209-019-02417-3