I am confused by the following corollary that appears in this book (just search "Corollary 7.8.11" for the appropriate page):
Corollary 7.8.11 Let $f : A \to B$ be a ring homomorphism and suppose that $B$ is finitely generated as a left $A$-module. Then if $A$ is left Noetherian or left Artinian, so is $B$
and am equally confused by this rather terse proof:
Every $B$-module is an $A$-module via $f$. So chain of ideals in $B$ is a chain $A$-submodules of a finitely generated $A$-module $B$.
First of all, in the statement of the collary, it says "Then if $A$ is left Noetherian or left Artinian, so is $B$" So is $B$ what? Are we trying to show $B$ is a left Noetherian ring or module?
"Every $B$-module is..." What's the purpose of this remark? Why do we care about $B$-modules? The statement of the corollary doesn't seem to indicate that we need to consider $B$-modules.Sure, if you have ring homomorphism $f : R \to S$ between rings $R$ and $S$, and $M$ is an $S$-module, then $M$ has a natural $R$-module structure given by $r \cdot m = f(r)m$. But in our case $B$ is already given some $A$-module structure already (indeed, it's finitely generated $A$-module). I don't see what role the module structure induced by $f$ is playing in the proof.
I'm hoping that someone will help me flesh out this proof. If it saves you any trouble, I'm really only interested in the Noetherian case, as the Artinian case is similar.