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a+b and ab are the usual integer addition and multiplication. You can assume that this new operation forms a ring, say R is the set of integers with these operations. Then does R have zero-divisors? And Are R and Z (the integers) isomorphic?

Any help will be appreciated!

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  • $\begingroup$ It is a terrible idea to split the question between title and body. Write the entire question in the body please. $\endgroup$ – rschwieb May 2 '16 at 23:08
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There's a simple isomorphism $$\varphi : R \to \mathbb{Z},~~z \mapsto 1-z$$ From this it immediately follows that $R$ does not have zero divisors, since $\mathbb{Z}$ does not either.

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  • $\begingroup$ Thank you! In addition, Are Z and R isomorphic? $\endgroup$ – Jordy Nelson May 3 '16 at 3:41
  • $\begingroup$ If an isomorphism exists, they're isomorphic. I just gave you an isomorphism. $\endgroup$ – Anon May 3 '16 at 7:05

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