Let $R$ be a (possibly noncommutative, but unital) ring. There are several finiteness conditions on a left $R$-module that we can consider. The ones I have come across include
- finitely generated
- finitely presented
- Noetherian
- Artinian
- finite length
There are several relations between these that hold in general. For instance
- finitely presented $\Rightarrow$ finitely generated
- Noetherian $\Rightarrow$ finitely generated (actually Noetherian $\Leftrightarrow$ every submodule is finitely generated)
- Noetherian and Artinian $\Leftrightarrow$ finite length
Yet the finiteness conditions above do not coincide in general. For example, there are rings which have Artinian but non-Noetherian left modules, etc.
However, if the ring $R$ itself satisfies some similar conditions for its (left) ideals, then some of the finiteness conditions coincide for left $R$-modules. I am interested in the following problem:
Given a set of finiteness conditions, classify the rings where these conditions coincide for left $R$-modules.
I guess the most elementary example of this is the following:
1) $R$ is left Noetherian $\Leftrightarrow$ "finitely generated $\equiv$ Noetherian" for left $R$-modules.
The nontrivial part of $\Rightarrow$ -finitely generated left modules over a left Noetherian ring are Noetherian- is proved in probably every introductory textbook in module theory and the other direction comes for free by considering $R$ as a left module over itself.
Perhaps less well known, we also have:
2) $R$ is left Noetherian $\Leftrightarrow$ "finitely generated $\equiv$ finitely presented" for left $R$-modules.
This can be shown by using the following lemma (this is stated in Matsumura's book Commutative Ring Theory):
Lemma: Let $R$ be any ring. Given a short exact sequence of left $R$-modules
$$0\longrightarrow M_1 {\longrightarrow}M_2{\longrightarrow}M_3\longrightarrow 0 $$
if $M_3$ is finitely presented and $M_2$ is finitely generated, then $M_1$ is finitely generated.
Proof. There is an exact sequence of the form
$$R^m {\longrightarrow}R^n{\longrightarrow}M_3\longrightarrow 0 $$ and as it involves projective modules, there are maps $f:R^m \rightarrow M_1$ and $g: R^n \rightarrow M_2$
such that together with the identity map $M_3 \rightarrow M_3$, they connect the above two exact sequences in a commutative diagram. Now by the snake lemma, we get that $\text{coker}\,f \cong \text{coker}\,g$ is finitely generated. $\text{im} \, f$ is also finitely generated, hence $M_1$ is finitely generated.
Proof of (2): $\Rightarrow$ is easy, we can surject on a finitely generated module by a finitely generated free module and the kernel will be finitely generated also by the Noetherian assumption. For $\Leftarrow$, let $I$ be a left ideal of $R$, then the above lemma applies for the short exact sequence $$0\longrightarrow I {\longrightarrow}R{\longrightarrow}R/I\longrightarrow 0 $$ and we get that $I$ is finitely generated.
$R$ is said to be semiprimary if its Jacobson radical $\text{rad} \, R$ is nilpotent and $R / \text{rad} \, R$ is semisimple. The Hopkins-Levitzki theorem stated as in T.Y. Lam's book$\,$ A First Course in Noncommutative Rings says that
3) $R$ is semiprimary $\Rightarrow$ "Noetherian $\equiv$ Artinian $\equiv$ finite length" for left $R$-modules.
I don't know whether (3) can be made into an if and only if. So I'd like to know of a counterexample or a proof to the reverse implication. Note that since being semiprimary is a left-right symmetric notion, (3) is also true for right $R$-modules. This makes me believe that the converse of (3) is not true.
Now if $R$ is left Artinian, it is semiprimary and hence left Noetherian by (3). Combining (1), (2) and (3) we see that all the five finiteness conditions I listed above coincide for left Artinian rings. Also this happens only for left Artinian rings by considering $_R R$. That is, we have
4) $R$ is left Artinian $\Leftrightarrow$ "finitely generated $\equiv$ finitely presented $\equiv$ Noetherian $\equiv$ Artinian $\equiv$ finite length" for left $R$-modules.
I'm most interested in knowing for exactly which rings the Noetherian and Artinian conditions coincide on left modules. But answers considering other interesting combinations of finiteness conditions are also welcome.
