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I have two binary operations, $ \bullet $ and $ \star $, defined on the abelian group $ (\Bbb{C} \oplus \Bbb{C},+) $ by $$ (a,b) \bullet (c,d) = (a c,b d) \quad \text{and} \quad (a,b) \star (c,d) = (a c + b d,a d + b c). $$ Each of these binary operations makes $ (\Bbb{C} \oplus \Bbb{C},+) $ a ring.

Question. Is it necessarily true that $ (\Bbb{C} \oplus \Bbb{C},+,\bullet) $ and $ (\Bbb{C} \oplus \Bbb{C},+,\star) $ are isomorphic rings? If so, what would be an explicit isomorphism between them?

Thanks for your help!

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  • $\begingroup$ In fact, I think this is true even if $ \Bbb{C} $ is replaced by any field $ \Bbb{F} $ of characteristic $ p > 2 $. $\endgroup$ May 3, 2015 at 22:19

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Hint: Consider the map $ (\Bbb{C} \oplus \Bbb{C},+,\star) \to (\Bbb{C} \oplus \Bbb{C},+,\bullet) $ defined by $ (a,b) \mapsto (a + b,a - b) $.

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