Abstract
LetA be an abelian group; we denote byd(A) the homological dimension ofA as a module over its ringE of endomorphisms. In the present note we exhibit a groupA for whichgl dimE=∞ andd(A)≤1, we show that there is a class of torsion-free groupsA of rank>2 for whichd(A)=∞ and we construct an indecomposable torsion-free group of rankr which isE-isomorphic toE.
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Douglas, A. J., andH. K. Farahat: The homological dimension of an Abelian groups as a module over its ring of endomorphisms. Mh. Math.69, 294–305 (1965).
Douglas, A. J., andH. K. Farahat: The homological dimension of an Abelian group as a module over its ring of endomorphisms. II. Mh. Math.76, 109–111 (1972).
L. Fuchs: Abelian groups. Pergamon. 1960.
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This research was carried out with the support of the National Research Council of Canada (Grant #A7259) during the summer of 1972.
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Douglas, A.J., Farahat, H.K. The homological dimension of an abelian group as a module over its ring of endomorphisms. III. Monatshefte für Mathematik 80, 37–44 (1975). https://doi.org/10.1007/BF01487802
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DOI: https://doi.org/10.1007/BF01487802