Zöschinger studied modules whose radicals have supplements and called these modules radical supplemented. Motivated by this, we call a module strongly radical supplemented (briefly srs) if every submodule containing the radical has a supplement. We prove that every (finitely generated) left module is an srs-module if and only if the ring is left (semi)perfect. Over a local Dedekind domain, srs-modules and radical supplemented modules coincide. Over a nonlocal Dedekind domain, an srs-module is the sum of its torsion submodule and the radical submodule.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
R. Wisbauer, Foundations of Modules and Rings, Gordon and Breach (1991).
H. Zöschinger, “Basis-Untermoduln und Quasi-kotorsions-Moduln über diskreten Bewertungsringen,” Bayer. Akad. Wiss. Math-Nat. Kl. Sitzungsber, 9–16 (1977).
H. Zöschinger, “Moduln, die in jeder Erweiterung ein Komplement haben,” Math. Scand., 35, 267–287 (1974).
H. Zöschinger, “Komplementierte moduln über Dedekindringen,” J. Algebra, 29, 42–56 (1974).
R. Alizade, G. Bilhan, and P. F. Smith, “Modules whose maximal submodules have supplements,” Commun. Algebra, 29, No. 6, 2389–2405 (1987).
J. Clark, C. Lomp, N. Vanaja, and R. Wisbauer, Lifting Modules. Supplements and Projectivity in Module Theory. Frontiers in Mathematics, Birkhäuser, Basel (2006).
E. Büyükaşık and C. Lomp, “Rings whose modules are weakly supplemented are perfect. Application to certain ring extension,” Math. Scand., 106, 25–30 (2009).
Author information
Authors and Affiliations
Additional information
Published in Ukrains’kyi Matematychnyi Zhurnal, Vol. 63, No. 8, pp. 1140–1146, August, 2011.
Rights and permissions
About this article
Cite this article
Büyükaşık, E., Türkmen, E. Strongly radical supplemented modules. Ukr Math J 63, 1306–1313 (2012). https://doi.org/10.1007/s11253-012-0579-3
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11253-012-0579-3