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Does every infinitely generated Abelian group admit a field structure? (i.e. If $(G,+)$ is an infinitely generated Abelian group, then, is there a binary operation "$\cdot$" such that $(G,+,\cdot)$ is a field?)
Is there any characterization (or classification) of such these groups (admitting field structure) ?
(What about Abelian groups admitting ring structure with nontrivial multiplication?)
A field is a vector space over its prime field, so its additive group is either a group of prime exponent $p$, or a torsion-free divisible group, depending on the characteristic. Conversely, any such abelian group is an additive group of an extension of $\mathbb F_p$ or $\mathbb Q$ of appropriate degree.
No. Consider for instance the direct product of all groups of distinct prime order.
This question is discussed already in MO and you can find the answer in the links below:
1- https://mathoverflow.net/questions/154852/a-basic-question-about-rings/154882#154882
3- https://mathoverflow.net/questions/155179/potentially-identity-elements-in-an-abelian-group
