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Can every abelian group converted into ring(by defining multiplication operation) with identity with same order. We can convert every group G into ring by defining a.b = 0 for all a and b in G. But this ring has no multiplicative identity

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The classification of finitely generated abelian groups implies that the group is a direct sum of copies of $\mathbb{Z}$ and copies of $\mathbb{Z}/n\mathbb{Z}$ for certain natural numbers $n \gt 1$. Endowing these groups with the natural ring structure coordinate-wise gives a non-trivial ring structure.

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    $\begingroup$ What about the non finitely generated case ? $\endgroup$
    – Amr
    Oct 17, 2021 at 19:41

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