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Show that $\mathbb{R}\oplus\mathbb{R}$ is not ring isomorphic to $\mathbb{C}$.

This is my first abstract algebra class and therefore if the explanation could be kept as simple as possible that would be very much appreciated. I dont even know how to approach this problem.

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The first thing I would think about is: is $\mathbf{R} \oplus \mathbf{R}$ even a field? –  Dylan Moreland Nov 28 '11 at 5:56
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When you want to show that two rings are not isomorphic, the usual approach is to find some property which is enjoyed by one of them and not by the other, and which is preserved by isomorphisms. For example, the properties «being a field» or, more simply, «being an integral domain» or «containing an element whose square is the additive inverse of $1$» do the trick here. –  Mariano Suárez-Alvarez Nov 28 '11 at 6:19

1 Answer 1

  • Nontrivial direct sums always have zero divisors, and $\mathbb C$ doesn't.
  • $\mathbb C$ has other algebraic properties that $\mathbb R\oplus \mathbb R$ doesn't have, like existence of solutions to polynomial equations, which would be preserved under ring isomorphism. For example, $\mathbb R\oplus \mathbb R$ has no element whose square is $(-1,-1)$, whereas $z^2=-1$ does have solutions in $\mathbb C$.
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