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On LCA groups whose ring of continuous endomorphisms satisfies DCC on closed ideals. (English) Zbl 1460.22002

Summary: We determine the structure of LCA (locally compact abelian) groups \(X\) with the property that the ring \(E(X)\) of continuous endomorphisms of \(X\) taken with the compact-open topology, satisfies \(DCC\) (descending chain condition) on different types of closed ideals.

MSC:

22B05 General properties and structure of LCA groups
16W80 Topological and ordered rings and modules
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[1] Armacost D. L. The structure of locally compact abelian groups. Pure and Applied Mathematics Series, Vol. 68 (Marcel Dekker, ed.), New York, 1981. · Zbl 0509.22003
[2] Armacost D. L. and Armacost W. L. On Q-dense and densely divisible LCA groups, Proc. Amer. Math. Sos., 1976, 36, 301-305. · Zbl 0255.22006
[3] Bourbaki N. Topologie generale, Chapter 1-2, ´El´ements de mathematique,Nauka, Moscow, 1968. · Zbl 0165.56403
[4] Bourbaki N. Topologie generale, Chapter 3-8, ´El´ements de mathematique,Nauka, Moscow, 1969.
[5] Bourbaki N. Topologie generale, Chapter 9-20, ´El´ements de mathematique.Nauka, Moscow, 1975.
[6] Fuchs L. Infinite abelian groups, Vol. 1. Academic Press, New York and London, 1970. · Zbl 0209.05503
[7] Fuchs L. Infinite abelian groups, Vol. 2. Academic Press, New York and London, 1973. · Zbl 0257.20035
[8] Hewitt E., Ross K. Abstract Harmonic Analysis, Vol. 1. Academic Press, New York, 1963. · Zbl 0115.10603
[9] Kert´esz A. Lectures on artinian rings,Acad´emiai Kiad´o, Budapest, 1987. · Zbl 0681.16001
[10] Lam T. Y. A first course in noncommutative rings, Graduate texts in mathematics, Vol. 131, Springer-Verlag, Berlin-Heidelberg-New York, 1991. · Zbl 0728.16001
[11] Popa V. Units, idempotents and nilpotents of an endomorphism ring. I, Bul. Acad. S¸tiint¸e Repub. Moldova, Mat., 1996, No. 3(22), 83-93. · Zbl 1114.16316
[12] Popa V. On LCA groups whose rings of continuous endomorphisms have at most two nontrivial closed ideals. I,Bul. Acad. S¸tiint¸e Repub. Moldova, Mat., 2011, No. 3(67), (2011), 91-107. · Zbl 1251.22004
[13] Robertson L. Connectivity, divisibility and torsion, Trans. Amer. Moath. Sos., 1967, 128, 482-505. · Zbl 0153.04401
[14] Rose H. Linear algebra: a pure mathematical approach, Birkh˝auser Verlag, Basel-BostonBerlin, 2002. · Zbl 1010.15001
[15] Sz´asz F. Die abelschen Gruppen, deren volle Endomorphismenringe die Minimalbedingung f˙ur Hauptrechtsideale erf ˙ullen, Monatshefte Math., 1961, 65, 150-153. · Zbl 0104.02504
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