Are the rings $\mathbb C[x,y]/\langle x^2+y^2-1 \rangle$ and $\mathbb C[x,y]/\langle xy-1 \rangle$ isomorphic ? I don't know whether any of the rings can be reduced to a simpler looking ring , and the way as it is , it is seemingly hard to find their zero divisors or idempotents . Please help
2026-03-25 23:39:23.1774481963
Are the rings $\mathbb C[x,y]/\langle x^2+y^2-1 \rangle$ and $\mathbb C[x,y]/\langle xy-1 \rangle$ isomorphic ?
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Hint $\,\ 1 = x^2+y^2 = (x-iy)(x+iy) = XY $