Finite rank torsion-free abelian groups uniserial over their endomorphism rings

Hausen, Jutta

doi:10.1090/S0002-9939-1985-0770526-1

Finite rank torsion-free abelian groups uniserial over their endomorphism rings
HTML articles powered by AMS MathViewer

by Jutta Hausen PDF

Proc. Amer. Math. Soc. 93 (1985), 227-231 Request permission

Abstract:

An abelian group is $E$-uniserial if its lattice of fully invariant subgroups is totally ordered. Finite rank torsion-free reduced $E$-uniserial groups are characterized. Such a group is a free module over the center $C$ of its endomorphism ring, and $C$ is a strongly indecomposable discrete valuation ring. Properties similar to those of strongly homogeneous groups are derived.

References

David M. Arnold, Strongly homogeneous torsion free abelian groups of finite rank, Proc. Amer. Math. Soc. 56 (1976), 67–72. MR 399305, DOI 10.1090/S0002-9939-1976-0399305-9
R. A. Bowshell and P. Schultz, Unital rings whose additive endomorphisms commute, Math. Ann. 228 (1977), no. 3, 197–214. MR 498691, DOI 10.1007/BF01420290
Shalom Feigelstock, The additive groups of rings with totally ordered lattice of ideals, Quaestiones Math. 4 (1980/81), no. 4, 331–335. MR 639432

Additive groups of rings

Jutta Hausen, Abelian groups which are uniserial as modules over their endomorphism rings, Abelian group theory (Honolulu, Hawaii, 1983) Lecture Notes in Math., vol. 1006, Springer, Berlin, 1983, pp. 204–208. MR 722619, DOI 10.1007/BFb0103703
Jutta Hausen, The additive group of rings with totally ordered ideal lattices, Quaestiones Math. 6 (1983), no. 4, 323–332 (1984). MR 734653
Irving Kaplansky, Infinite abelian groups, Revised edition, University of Michigan Press, Ann Arbor, Mich., 1969. MR 0233887
James D. Reid, On the ring of quasi-endomorphisms of a torsion-free group, Topics in Abelian Groups (Proc. Sympos., New Mexico State Univ., 1962), Scott, Foresman & Co., Chicago, Ill., 1963, pp. 51–68. MR 0169915
J. D. Reid, Abelian groups finitely generated over their endomorphism rings, Abelian group theory (Oberwolfach, 1981) Lecture Notes in Math., vol. 874, Springer, Berlin-New York, 1981, pp. 41–52. MR 645915
P. Schultz, The endomorphism ring of the additive group of a ring, J. Austral. Math. Soc. 15 (1973), 60–69. MR 0338218