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Now ℝxℝ≅ℂ, seen by the map that sends (a,b) to a + bi. ℂ is a field, so the product of any two non-zero elements is non-zero. However, this doesn't seem to hold in ℝxℝ, since (1,0) * (0,1) = (0,0) even though (1,0) and (0,1) are non-zero. What am I missing?
Also, the zero ideal is maximal in ℂ, since ℂ is a field. But in ℝxℝ, we have two maximal ideals: ℝx{0} and {0}xℝ. Surely the isomorphism between ℝxℝ and ℂ ought to preserve the ideal structure?
Also, the zero ideal is maximal in ℂ, since ℂ is a field. But in ℝxℝ, we have two maximal ideals: ℝx{0} and {0}xℝ. Surely the isomorphism between ℝxℝ and ℂ ought to preserve the ideal structure?