Finite rank torsionfree abelian groups uniserial over their endomorphism rings
Author:
Jutta Hausen
Journal:
Proc. Amer. Math. Soc. 93 (1985), 227231
MSC:
Primary 20K15
MathSciNet review:
770526
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Abstract: An abelian group is uniserial if its lattice of fully invariant subgroups is totally ordered. Finite rank torsionfree reduced uniserial groups are characterized. Such a group is a free module over the center of its endomorphism ring, and is a strongly indecomposable discrete valuation ring. Properties similar to those of strongly homogeneous groups are derived.
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 R. A. Bowshell and P. Schultz, Unital rings whose additive endomorphisms commute, Math. Ann. 228 (1977), 197214. MR 0498691 (58:16768)
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 S. Feigelstock, The additive groups of rings with totally ordered lattice of ideals, Quaestiones Math. 4 (1981), 331335. MR 639432 (83b:17003)
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 , Additive groups of rings, Research Notes in Math., vol. 83, Pitman, Boston, Mass., 1983.
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 J. Hausen, Abelian groups which are uniserial as modules over their endomorphism rings, Abelian Group Theory, Lecture Notes in Math., vol. 1006, SpringerVerlag, Berlin and New York, pp. 204208. MR 722619 (85c:20048)
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 , The additive group of rings with totally ordered ideal lattices, Quaestiones Math. 6 (1983), 323332. MR 734653 (85i:20061)
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 I. Kaplansky, Infinite abelian groups, rev. ed., Univ. of Michigan Press, Ann Arbor, 1969. MR 0233887 (38:2208)
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 J. D. Reid, On the ring of quasiendomorphisms of a torsionfree group, Topics in Abelian Groups, Scott, Foresman, Chicago, 1963, pp. 5168. MR 0169915 (30:158)
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 , Abelian groups finitely generated over their endomorphism rings, Abelian Group Theory, Lecture Notes in Math., vol. 874, SpringerVerlag, Berlin and New York, 1981, pp. 4152. MR 645915 (83e:20061)
 [10]
 P. Schultz, The endomorphism ring of the additive group of a ring, J. Austral. Math. Soc. 15 (1973), 6069. MR 0338218 (49:2984)
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Additional Information
DOI:
http://dx.doi.org/10.1090/S00029939198507705261
PII:
S 00029939(1985)07705261
Keywords:
uniserial,
torsionfree abelian group,
valuation domain
Article copyright:
© Copyright 1985
American Mathematical Society
