Abstract
If \(G\) is a Hopfian abelian group then it is, in general, difficult to determine if direct sums of copies of \(G\) will remain Hopfian. We exhibit large classes of Hopfian groups such that every finite direct sum of copies of the group is Hopfian. We also show that for any integer \(n > 1\) there is a torsion-free Hopfian group \(G\) having the property that the direct sum of \(n\) copies of \(G\) is not Hopfian but the direct sum of any lesser number of copies is Hopfian.
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References
R.A. Beaumont, R.S. Pierce, Isomorphic direct summands of abelian groups. Math. Ann. 153, 21–37 (1964)
P.M. Cohn, Some remarks on the invariant basis property. Topology 5, 215–228 (1966)
A.L.S. Corner, Every countable reduced torsion-free ring is an endomorphism ring. Proc. Lond. Math. Soc. 13, 687–710 (1963)
A.L.S. Corner, Three examples on hopficity in torsion-free abelian groups. Acta Math. Acad. Sci. Hungar. 16, 303–310 (1965)
A.L.S. Corner, R. Göbel, Prescribibg endomorphism algebras: a unified treatment. Proc. Lond. Math. Soc. 50, 447–479 (1985)
L. Fuchs, Infinite Abelian Groups, vol. I (Academic Press, New York, 1970)
L. Fuchs, Infinite Abelian Groups, vol. II (Academic Press, New York, 1973)
B. Goldsmith, Anthony Leonard Southern Corner 1934–2006, in Models, Modules and Abelian Groups, ed. by R. Göbel, B. Goldsmith (Walter de Gruyter, Berlin, 2008), pp. 1–7
B. Goldsmith, K. Gong, A note on hopfian and co-hopfian abelian groups. Contemp. Math. 576, 129–136 (2012)
B. Goldsmith, L. Strüngmann, Torsion-free weakly transitive abelian group. Comm. Algebra 33, 1171–1191 (2005)
K.R. Goodearl, von Neumann Regular Rings (Pitman (Advanced Publishing Program), Boston, 1979)
K.R. Goodearl, Surjective endomorphisms of finitely generated modules. Comm. Algebra 15, 589–609 (1987)
I. Kaplansky, Modules over Dedekind rings and valuation rings. Trans. Am. Math. Soc. 72, 327–340 (1952)
I. Kaplansky, Infinite Abelian Groups (University of Michigan Press, Ann Arbour, 1954 and 1969)
J.C. McConnell, J.C. Robson, Noncommutative Noetherian Rings, revised ed., Graduate Studies in Mathematics, vol. 30 (American Mathematical Society, Providence, RI, 2001) (With the cooperation of L. W. Small)
N.H. McCoy, Divisors of zero in matric rings. Bull. Am. Math. Soc 47, 166–172 (1941)
P.M. Neumann, Pathology in the Representation Theory of Infinite Soluble Groups, in Proceedings of ‘ Groups-Korea 1988’. Lecture Notes in Mathematics 1398 (Eds A.C. Kim and B.H. Neumann), pp. 124–139
R.S. Pierce, Homomorphisms of Primary Abelian Groups, Topics in Abelian Groups (Scott Foresman, Chicago, 1963), pp. 215–310
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Goldsmith, B., Vámos, P. The Hopfian exponent of an abelian group. Period Math Hung 69, 21–31 (2014). https://doi.org/10.1007/s10998-014-0038-z
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DOI: https://doi.org/10.1007/s10998-014-0038-z