I am looking for an answer to why one has to assume commutativity of a ring $R$ in proving some results about Noetherian rings. For example, Let $R$ be a commutative ring; look at the proof(s) of the theorem "if $H$ and $R/H$ are Noetherian, then so is $R$". Where does commutativity comes in?
EDIT
For example, consider the following sketch of a proof. Consider a generic ascending chain of ideals $\{H_i\}_i$ in $R$. We know (by hypothesis) that $\{H_i \cap H\}_i$ is stationary so is $\{(H_i + H)/H\}_i$. Pick the max index that makes both chains stationary (say $m$), then using Dedekind's law for groups starting from $H_i = H_i\cap(H_i+H)$ (note that it is $H \cap H_i = H \cap H_m$ and $H+H_i =H+H_m$ from $m$ on), we finally get that $\{H_i\}_i$ is stationary.