标题： 关于有限生成阿贝尔群的Fuchs问题：小扭子情况 显示英文标题

标题： On Fuchs' problem for finitely generated abelian groups: The small torsion case

摘要： 一个经典的难题，由Fuchs在1960年提出，要求对某些环的单位群进行分类。 在本文中，我们考虑有限生成阿贝尔群的情况，在额外假设挠子群较小的情况下解决了Fuchs的问题，这里的“较小”是与Prüfer秩相关的适当定义。 作为一个具体实例，我们对每个$n\ge2$对形如$\mathbb{Z}/n\mathbb{Z}\times\mathbb{Z}^r$的可实现群进行分类。 我们的工具需要对奇素数幂阶的适当根环的伴随群进行研究，给出加法群和伴随群同构的条件。 在最后一节中，我们也处理了一些阶为$2$幂的群，证明形如$\mathbb{Z}/4\mathbb{Z}\times \mathbb{Z}/2^{u}\mathbb{Z}$的群是可实现的当且仅当$0\le u\le 3$或$2^u+1$是费马素数。 显示英文摘要

摘要： A classical problem, raised by Fuchs in 1960, asks to classify the abelian groups which are groups of units of some rings. In this paper, we consider the case of finitely generated abelian groups, solving Fuchs' problem for such group with the additional assumption that the torsion subgroups are small, for a suitable notion of small related to the Pr\"ufer rank. As a concrete instance, we classify for each $n\ge2$ the realisable groups of the form $\mathbb{Z}/n\mathbb{Z}\times\mathbb{Z}^r$. 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 $\mathbb{Z}/4\mathbb{Z}\times \mathbb{Z}/2^{u}\mathbb{Z}$ are realisable if and only if $0\le u\le 3$ or $2^u+1$ is a Fermat's prime.