# Morita contexts and Noetherianity/affineness

Let $(R\,,\, S\,,\, _RM_S\,,\, _SN_R\,,\, f\,,\, g)$ be a Morita context with $NM=S$ and $R$ right Noetherian. Show that $S$ is right Noetherian as well. If we further assume $R$ is an affine $\mathbb{k}$-algebra (for some commutative ring $\mathbb{k}$), show that $S$ is as well.

I've tried to mimic a bit of what I've seen in some texts I've skimmed through, but it hasn't helped so far. I've been trying to find an inclusion of the lattice of submodules of $S$ into the lattice of submodules of $R$; perhaps it's better to try and do it directly.... But I don't know how to relate the Morita context to a chain of ideals of $S$.

Any help is appreciated

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When is a ring "k affine"? –  rschwieb Apr 15 '13 at 21:18
Whoops! I meant to assume that $R$ is an affine $\mathbb{k}$-algebra. It has been edited –  Bey Apr 16 '13 at 11:26
I have no idea what you mean by "affine algebra" either. Wikipedia seems to indicate you might mean "finitely generated algebra"? –  rschwieb Apr 16 '13 at 13:15
Yes, "affine" means it is finitely generated as an algebra over $\mathbb{k}$. Sorry for not being clear; I thought it conventional terminology –  Bey Apr 16 '13 at 18:53
So for instance, you consider $F[x]$ is an affine $F$ algebra, right? Hmm, that's interesting: I do not ever recall hearing that being an affine $k$ algebra is Morita invariant, although it wouldn't surprise me. Does this property have a module theoretic characterization? –  rschwieb Apr 16 '13 at 19:30

Probably the way to think of it is to establish the functor $F(-):Mod-R\to Mod-S$ isomorphic to $-\otimes_R M$ for a progenerator $_RM_S$ and then look at the properties it preserves.
First establish that if $N_R$ is finitely generated, then so is the left $S$ module $F(N)$. Then, identifying the submodules of $F(N)$ as images of submodules of $N$ ia $F$, you can easily say that all submodules of $F(N)$ are finitely generated.
Applying this to $N_R=R_R$ would finish the job.
Actually it is probably possible to argue directly with a correspondence of submodules using $F$ and a counterpart inverse functor $G$, but I lack the experience to state that confidently.