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Finite rank torsion-free abelian groups uniserial over their endomorphism rings
| Content Provider | Semantic Scholar |
|---|---|
| Author | Hausen, Jutta |
| Copyright Year | 1985 |
| Abstract | An abeian 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. The results. In the sequel all groups will be abelian. This paper was motivated by the recent study of additive groups of valuation rings (i.e. rings whose lattice of two-sided ideals forms a chain) [3, 4, 6]. Obviously, the additive group of a valuation ring is uniserial regarded as a module over its endomorphism ring. We call such groups E-uniserial for short. In [5] we characterized the E-uniserial groups up to torsion-free reduced direct summands. Combining the results of [5] with Theorem 1 below yields the structure of all E-uniserial groups of finite torsion-free rank. We shall prove THEOREM 1. Let G be a torsion-free reduced abelian group of finite rank. Then the following conditions are equivalent. (a) G is E-uniserial. (b) G Hm where H is a strongly indecomposable E-uniserial group. (c) G is a free module over a valuation ring. (d) The center C = Cent E(G) of the endomorphism ring of G is a strongly indecomposable discrete valuation E-ring, and G is a free C-module. (e) E(G) Mat,m(C) where C is a discrete valuation ring and rank G = m (rank C). We use the phrase "discrete valuation ring" in the sense of Kaplansky [7, p. 42]. In particular, a discrete valuation ring is a principal ideal domain. A (discrete) valuation E-ring is a (discrete) vaiuation ring which is also an E-ring. E-rings were introduced by P. Schultz [10] as the rings whose additive endomorphisms are given by left multiplication with elements. Throughout, E(G) is the endomorphism ring of G, the center of a ring R is denoted by Cent R, and Mat m(R) is the ring of m x m matrices with entries in R. Received by the editors February 16, 1984. 1980 Mathematics Subject Classification. Primary 20K15. |
| Starting Page | 227 |
| Ending Page | 231 |
| Page Count | 5 |
| File Format | PDF HTM / HTML |
| DOI | 10.1090/S0002-9939-1985-0770526-1 |
| Volume Number | 93 |
| Alternate Webpage(s) | https://www.ams.org/journals/proc/1985-093-02/S0002-9939-1985-0770526-1/S0002-9939-1985-0770526-1.pdf |
| Alternate Webpage(s) | https://doi.org/10.1090/S0002-9939-1985-0770526-1 |
| Language | English |
| Access Restriction | Open |
| Content Type | Text |
| Resource Type | Article |
