The Hopfian exponent of an abelian group
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The Hopfian exponent of an abelian group Brendan Goldsmith · Peter Vámos
© Akadémiai Kiadó, Budapest, Hungary 2014
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. Keywords
Abelian groups · Hopfian groups · Leavitt rings
Mathematics Subject Classification
Primary 20K30 · Secondary 20K20 · 20K10
1 Introduction All rings in this paper will have identity elements, all modules will be unitary with the ring acting on the right; homomorphisms will act and will be written on the left. Recall that a module G is said to be Hopfian if every surjection f : G G is an automorphism; equivalently G is Hopfian if, and only if, it has no proper isomorphic factor. We obtain the dual notion of co-Hopfian by specifying that every injective endomorphism should be an automorphism. These concepts apply to the class of arbitrary groups (in fact, to universal algebras as well) but here we shall restrict our attention to modules and in particular to abelian groups. In fact, throughout this paper a group will mean an additively written abelian group. It is clear that a direct summand of a Hopfian module is again Hopfian and an infinite direct sum of copies of a non-zero module is never Hopfian. However, finite direct sums of copies of a module may exhibit some unexpected subtle behaviour: Corner in [4] showed the B. Goldsmith (B) School of Mathematical Sciences, Dublin Institute of Technology, Kevin Street, Dublin 8, Ireland e-mail: [email protected] P. Vámos Department of Mathematics, University of Exeter, Exeter, UK e-mail: [email protected]
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existence of a torsion-free group G which is Hopfian but G ⊕ G is not. Motivated by this example we introduce the following notion: Definition 1.1 The Hopfian exponent of a module G, denoted by Hex(G), is the the least natural number k > 0 for which the direct sum of k copies of G, denoted by G k , is not Hopfian. If no such k exists then Hex G = ∞. Thus a module is not Hopfian precisely when its Hopfian exponent is 1, while Hopfian groups have exponent > 1, so Corner’s example in [4] has Hopfian exponent precisely 2. Using the additive function of torsion-free rank we see that a torsion-free group of finite rank is always Hopfian—see for example, [9]. It follows that the group Z has Hex(Z) = ∞. Further, it is routine to see that a Noetherian module is Hopfian hence has Hopfian exponent ∞. It follows from this that if a ring is the union of its right Noetherian subrings (e.g. a commutative ring) then every finitely generated module has Hopfian exponent ∞. In fact, Goodearl in [12] completely characterized rings for which every finitely generated
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