# Are these ring structures on $(\Bbb{C} \oplus \Bbb{C},+)$ isomorphic?

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!

• In fact, I think this is true even if $\Bbb{C}$ is replaced by any field $\Bbb{F}$ of characteristic $p > 2$. – Berrick Caleb Fillmore May 3 '15 at 22:19

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