Abstract
Socles, radicals, and socle-radicals have long been an object of study in abelian groups (and in modules). In this paper we consider the “building blocks” for socles and radicals — those determined by a single group. These are called singly generated. This paper addresses the question of when these singly generated socles (radicals) are also radicals (socles). This problem is equivalent to the question of when certain classes (determined by a single group) are closed under extensions.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
B. Charles, “Sous-groupes fonctoriels et topologies”, Studies on Abelian Groups, 75–92. (Paris, 1968 ).
S.U. Chase, “On group extensions and a problem of J.H.C. Whitehead”, Topics in Abelian Groups, 173–193. ( Chicago, Illinois, 1963 ).
T.H. Fay, E.P. Oxford, G.L. Walls, “Preradicals in abelian groups”, Houston J. Math., 8 (1982) 39–52.
T.H. Fay, E.P. Oxford, G.L. Walls, “Preradicals induced by homomorphisms”, (to appear).
T.H. Fay, E.P. Oxford, G.L. Walls, “Stabilizer and annihilator classes”, (to appear).
T.H. Fay, E.P. Oxford, G.L. Walls, “The lattice of preradicals”, (to appear).
L. Fuchs, Infinite Abelian Groups, Vols. I, II, Academic Press, (New York and London, 1970, 1973 ).
R. Göbel, “On Stout and Slender Groups”, J of Alg, 35, 39–55 (1975).
E.M. Martinez and L. Salce, “A homological characterization of certain subgroups of abelian p-groups”, Acta Math. Acad. Sci. Hungar. 39, (1–2) (1982), 143–147.
R. Mines, “Radicals and torsion-free groups”, Seminar notes, New Mexico State University, 1971.
R. Nunke, “Purity and subfunctors of the identity”, Topics in Abelian Groups, 121–171, ( Chicago, Illinois, 1963 ).
R. Nunke, “On the structure of Tor. I”, Proc. Colloq. Abelian Groups, 115–124 (Budapest, 1964); II Pac. J. Math 22 (1967), 453–464.
S.V. Ryckov, “On direct products of abelian groups”, Dokl. Akad. Nauk SSSR 252 no. 2 (1980), 301–302 (Russian). (English translation: Soviet Math. Doklady no. 3, 21 (1980), 747–748 (1981).
L. Salce, “Cotorsion Theories for Abelian Groups”, Sym. Math, 23 (Conf. Ab. Gr. and their Relationship to the theory of Modules INDAM, Rome, 1977 ) 11–23, Acad. Press, London, 1979.
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 1983 Springer-Verlag Berlin Heidelberg
About this chapter
Cite this chapter
Fay, T.H., Oxford, E.P., Walls, G.L. (1983). Singly Generated Socles and Radicals. In: Göbel, R., Lady, L., Mader, A. (eds) Abelian Group Theory. Lecture Notes in Mathematics, vol 1006. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-21560-9_48
Download citation
DOI: https://doi.org/10.1007/978-3-662-21560-9_48
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-12335-4
Online ISBN: 978-3-662-21560-9
eBook Packages: Springer Book Archive