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I can't think of a ring without unity that has a subring with unity. There must be some element in the parent ring that doesn't work with the subring's identity, but I'm struggling to see how that would be possible. Any suggestions?

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Hint: $\mathbb Z\times2\mathbb Z$

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  • $\begingroup$ I see that it has no identity. I don't see the subring without unity... some $H \times G $ such that $H \subset \mathbb{Z}, G \subset 2 \mathbb{Z}$? One of the elements in the parent ring must be the identity for the subring, but it doesn't contain (1,1) so I'm not sure how? $\endgroup$ – bkaiser Apr 24 '15 at 15:06
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    $\begingroup$ @caesar $\mathbb Z\times\{0\}$ is a ring with unity. It's unit obviously is not a unit for the whole ring $\mathbb Z\times2\mathbb Z$. $\endgroup$ – Thomas Andrews Apr 24 '15 at 15:11
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Any ring without a unit contains the zero ring, which is a ring with unity.

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