# Is this a surjection of rings? What am I doing wrong?


What am I doing wrong?

• I would suggest drawing some pictures. – David E Speyer Aug 29 '12 at 13:29
• That's what I did, it's where the confusion came from. – user38830 Aug 29 '12 at 13:46
• However - I drew the wrong picture. I embedded $T$ as $T\times\{1\}$, but it has to be embedded diagonally - with that in mind, everything becomes clear. – user38830 Aug 29 '12 at 14:44

The commutativity of maps $\omega = \textrm{pr}_{Z} \circ \kappa$ only implies that $\textrm{pr}_{Z}^{-1}(\omega(T)) \supset \kappa(T)$. The equality holds iff $\textrm{pr}_{Z}$ is injective, which is not the case.
• Ahh! That is, of course, true. I now also realized why this is not confusing (for anyone still confused): $\kappa$ embeds $T$ diagonally, which is why it becomes closed in $Z\times T$. – user38830 Aug 29 '12 at 14:43