Aclassical problem, raised by Fuchs in 1960, asks to classify the abelian groupswhich are groups of units of some rings. In this paper, we consider the case of finitely generated abelian groups, solving Fuchs’ problem for such groups with the additional assumption that the torsion subgroups are small, for a suitable notion of small related to the Prüfer rank. As a concrete instance, we classify for each 𝑛 ⩾ 2 the realisable groups of the form ℤ∕𝑛ℤ × ℤ𝑟. Our tools require an investigation of the adjoint group of suitable radical rings of odd prime power order appearing in the picture, giving conditions under which the additive and adjoint groups are isomorphic. In the last section, we also deal with some groups of order a power of 2, proving that the groups of the form ℤ∕4ℤ × ℤ∕2𝑢ℤ are realisable if and only if 0 ⩽ 𝑢 ⩽ 3 or 2𝑢 + 1is a Fermat prime.
On Fuchs’ problem for finitely generated abelian groups: The small torsion case
I. Del Corso
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2024-01-01
Abstract
Aclassical problem, raised by Fuchs in 1960, asks to classify the abelian groupswhich are groups of units of some rings. In this paper, we consider the case of finitely generated abelian groups, solving Fuchs’ problem for such groups with the additional assumption that the torsion subgroups are small, for a suitable notion of small related to the Prüfer rank. As a concrete instance, we classify for each 𝑛 ⩾ 2 the realisable groups of the form ℤ∕𝑛ℤ × ℤ𝑟. Our tools require an investigation of the adjoint group of suitable radical rings of odd prime power order appearing in the picture, giving conditions under which the additive and adjoint groups are isomorphic. In the last section, we also deal with some groups of order a power of 2, proving that the groups of the form ℤ∕4ℤ × ℤ∕2𝑢ℤ are realisable if and only if 0 ⩽ 𝑢 ⩽ 3 or 2𝑢 + 1is a Fermat prime.I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.
