I am having a hard time writing a bijective map between the two rings: $ R = \dfrac{k[x,y,z,u,v]}{\left<(x-y)z+uv\right>} \cong \dfrac{k[x,y,z,u,v]}{\left<(x-y)z\right>} = S. $
I know that we need to explicitly write down where the generators go from left to right and then from right to left.
Any help is greatly appreciated.
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Note: in one direction, we could map $k[x,y,z,u,v]\rightarrow k[x,y,z,u,v]/\left< (x-y)z\right>$, with $ x \mapsto x, y \mapsto y, z\mapsto z, $ but where would $u$ and $v$ go? Would $u\mapsto x-y$ and $v\mapsto -z$?
As for the other direction, $k[x,y,z,u,v]\rightarrow k[x,y,z,u,v]/\left< (x-y)z+uv\right>$, do we have $ x\mapsto x, y\mapsto y, z\mapsto z, u\mapsto \not 1, v\mapsto \not 1? $