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On the lattice of right ideals of the endomorphism ring of an abelian group

Published online by Cambridge University Press:  17 April 2009

Theodore G. Faticoni
Theodore G. Faticoni
Affiliation:
Department of Mathematics, Fordham University, Bronx NY. 10458, United States of America.
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Abstract

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Let A be an abelian group, let ∧ = End (A), and assume that A is a flat left ∧-module. Then σ = { right ideals I ⊂ ∧ | IA = A} generates a linear topology oil ∧. We prove that Hom(A,·) is an equivalence from the category of those groups B ⊂ An satisfying B = Hom(A, B)A, onto the category of σ-closed submodules of finitely generated free right ∧-modules. Applications classify the right ideal structure of A, and classify torsion-free groups A of finite rank which are (nearly) isomorphic to each A-generated subgroup of finite index in A.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 38 , Issue 2 , October 1988 , pp. 273 - 291
Copyright
Copyright © Australian Mathematical Society 1988

References

[1]Albreclit, U., ‘Baer's lemma and Fuch's problem 84a’, Trans. Amer. Math. Soc. 293 (1986), 565–582.Google Scholar
[2]Arnold, D.M., Finite Rank and Torsion-free Abelian Groups and Rings, Lecture Notes in Mathematics 931 (Springer-Verlag, New York, 1982).CrossRefGoogle Scholar
[3]Arnold, D.M., ‘Endomorphism rings and subgroups of finite rank torsion-free abelian groups’, Rocky Mountain J. Math. 12 (1982), 241–256.Google Scholar
[4]Arnold, D.M., Genera and Direct Sum Decompositions of Torsion-free Modules, Lecture Notes in Mathematics 616, pp. 197–218 (Springer-Verlag, New York, 1976).Google Scholar
[5]Arnold, D.M. and Lady, E.L., ‘Endomorphism rings and direct sums of torsion-free abelian groups’, Trans. Amer. Math. Soc. 211 (1975), 225–237.CrossRefGoogle Scholar
[6]Arnold, D.M. and Murley, C.E., ‘Abelian groups A, such that Hom(A, –) preserves direct sums of copies of A’, Pacific J. Math. 56 (1975), 7–20.CrossRefGoogle Scholar
[7]Camillo, V. and Fuller, K.R., ‘Rings whose faithful modules are flat’, Archiv. Math (Basel) 27 (1976), 522–525.Google Scholar
[8]Faticoni, T.G. and Goeters, P., ‘Examples of torsion-free abelian groups flat as modules over thier endomorplusm ring’, Proc. London Math. Soc. (to appear).Google Scholar
[9]Faticoni, T.G. and Goeters, P., ‘On torsion-free Ext’, Comm. Alg. (1988) (to appear).CrossRefGoogle Scholar
[10]Faticoni, T.G., ‘Semi-local localizations of rings and subdirect decompositions of modules’, J. Pure Appl. Alg. 46 (1987), 137–163.CrossRefGoogle Scholar
[11]Fuchs, L., Infinite Abelian Groups,: Volumes I and II (Academic Press, New York London, 1970, 1973).Google Scholar
[12]Hausen, J., ‘Modules with time sumnnmand intersection property’, (preprint).Google Scholar
[13]Huber, M. and Warfield, R.B. Jr., Homomorphisms between Cartesian Powers of an Abelian Group, Lecture Notes in Mathematics 874, pp. 202–227 (Springer-Verlag, New York, 1981).Google Scholar
[14]Lady, L., ‘Nearly isomorphic torsion-free abelian groups’, J. Algebra 35 (1975), 235–238.CrossRefGoogle Scholar
[15]Reid, J., ‘On subcommnutative rings’, Acta. Math. Acad. Sci. Hungar. 16 (1965), 23–26.Google Scholar
[16]Reid, J., ‘On the ring of quasi-endomorphisms of a torsion-free group’, in Topics in Abelian Groups, pp. 51–68 (Chicago, Illinois, 1963).Google Scholar
[17]Reiner, I., Maximal Orders (Academic Press, New York, 1975).Google Scholar
[18]Richman, F. and Walker, E.A., ‘Primary abelian groups as modules over their endomorphism rings’, Math Z 89 ((1965)), 77–81.Google Scholar
[19]Stenström, B., Rings of Quotients (Springer-Verlag, Berlin, Heidelberg, New York, 1975).Google Scholar