Q:

Ring isomorphisms maintain noetherian property? 

If a noetherian ring R is isomorphic to a ring S is S noetherian too? 
I am pretty sure that it isn't but can't find a counterexample
Thanks! 

A:

It is indeed.  Suppose $\phi:R\to S$ is an isomorphism.  If $P_1\subset P_2\subset\cdots$ is an ascending chain of ideals in $S$, then $\phi^{-1}(P_1)\subset\phi^{-1}(P_2)\subset\cdots$ is an ascending chain of ideals in $R$, and $P_n=P_{n+1}$ if and only if $\phi^{-1}(P_n)=\phi^{-1}(P_{n+1})$.

