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Does every Abelian group admit a ring structure?

How might I construct an abelian group that cannot be represented by any ring with its addition operation? Wikipedia suggests 3 related topics, but they are quite unfamiliar. Perhaps it is possible to construct simple examples?

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    If $(n_j)_{j=1}^\infty$ is a sequence of positive integers, then $\mathbb Z_{n_1}\oplus \mathbb Z_{n_2}\oplus\mathbb Z_{n_3}\oplus\cdots$ is the abelian group of a ring with identity if and only if $(n_j)$ is bounded.2012-02-17

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