Czechoslovak Mathematical Journal, Vol. 67, No. 2, pp. 367-377, 2017
Regularly weakly based modules over right perfect rings and Dedekind domains
Michal Hrbek, Pavel Růžička
Received November 23, 2015. First published March 1, 2017.
Abstract: A weak basis of a module is a generating set of the module minimal with respect to inclusion. A module is said to be regularly weakly based provided that each of its generating sets contains a weak basis. We study (1) rings over which all modules are regularly weakly based, refining results of Nashier and Nichols, and (2) regularly weakly based modules over Dedekind domains.
Keywords: weak basis; regularly weakly based ring; Dedekind domain; perfect ring
References: [1] F. W. Anderson, K. R. Fuller: Rings and Categories of Modules. Graduate Texts in Mathematics 13, Springer, New York (1992). DOI 10.1007/978-1-4612-4418-9 | MR 1245487 | Zbl 0765.16001 [2] A. J. Berrick, M. E. Keating: An Introduction to Rings and Modules with $K$-Theory in View. Cambridge Studies in Advanced Mathematics 65, Cambridge University Press, Cambridge (2000). MR 1757884 | Zbl 0949.16001 [3] V. Dlab: On a characterization of primary abelian groups of bounded order. J. Lond. Math. Soc. 36 (1961), 139-144. DOI 10.1112/jlms/s1-36.1.139 | MR 0123604 | Zbl 0104.02601 [4] D. Herden, M. Hrbek, P. Růžička: On the existence of weak bases for vector spaces. Linear Algebra Appl. 501 (2016), 98-111. DOI 10.1016/j.laa.2016.03.001 | MR 3485061 | Zbl 1338.15004 [5] M. Hrbek, P. Růžička: Weakly based modules over Dedekind domains. J. Algebra 399 (2014), 251-268. DOI 10.1016/j.jalgebra.2013.09.031 | MR 3144587 | Zbl 1308.13013 [6] M. Hrbek, P. Růžička: Characterization of Abelian groups with a minimal generating set. Quaest. Math. 38 (2015), 103-120. DOI 10.2989/16073606.2014.981704 | MR 3334638 [7] I. Kaplansky: Infinite Abelian Groups. University of Michigan Publications in Mathematics 2, University of Michigan Press, Ann Arbor (1954). MR 0065561 | Zbl 0057.01901 [8] P. A. Krylov, A. A. Tuganbaev: Modules over Discrete Valuation Domains. De Gruyter Expositions in Mathematics 43, Walter de Gruyter, Berlin (2008). DOI 10.1515/9783110205787 | MR 2387130 | Zbl 1144.13001 [9] B. Nashier, W. Nichols: A note on perfect rings. Manuscr. Math. 70 (1991), 307-310. DOI 10.1007/BF02568380 | MR 1089066 | Zbl 0721.16009 [10] J. Neggers: Cyclic rings. Rev. Un. Mat. Argentina 28 (1977), 108-114. MR 0463238 | Zbl 0371.16011 [11] D. S. Passman: A Course in Ring Theory. Wadsworth & Brooks/Cole Mathematics Series, Wadsworth & Brooks/Cole Advanced Books & Software, Pacific Grove (1991). MR 1096302 | Zbl 0783.16001
Affiliations: Michal Hrbek, Pavel Růžička, Department of Algebra, Faculty of Mathematics and Physics, Charles University, Sokolovská 83, 186 75 Praha, Czech Republic, e-mail: hrbmich@gmail.com, ruzicka@karlin.mff.cuni.cz
