TITLE: Infinitely Noetherian Rings
QUESTION [17 upvotes]: One equivalent definition of a noetherian ring is that it satisfies the ascending chain condition.  That is, for every ascending chain of ideals 
$$ I_{0} \subseteq I_{1} \subseteq I_{2} \subseteq \dots \subseteq I_{k} \subseteq I_{k+1} \dots $$
there is a j such that for all $k \ge j$, $I_{k} = I_{k+1}$.
What if this definition is extended into the ordinals?  Could there be a ring which satisfies the $j = \omega^2 + \omega\cdot3 + 5$ noetherian condition, for example? That is, could there be an infinite $j$ such that there are no ascending chains of length greater than $j$.  In this example, we would say that the ring is $\omega^2 +\omega\cdot3 + 5$ noetherian.  
Most importantly, what kind of rings would be created? Same question with a variety of definitions of noetherian.  
Edit:  The way I am trying to define such rings seems to be problematic.  I'd like to conceive a clearer definition, but I need to become a bit more familiar with noetherian rings.

REPLY [6 votes]: Possibly this is already too far away to qualify as 'answer', but it is long for a comment and might be of interest. 
There is a classical construction of descending (not ascending), so rather matching artinian than noetherian, subgroups of abelian groups (or, to artificial stay a bit closer let's say submodules of Z-modules, but then this got generalized to modules over other rings so it is not totally artificial), where indexing by ordinals is done and relevant. They are called Ulm subgroups. 
Let $G$ be an abelian group and $p$ some (fixed) prime. Than the (classical) p-height of an element $g$ is the supremum of all naturals such that the  equation $p^nx = g$ has a solution $x$ in $G$. 
One can express this differently by defining 
$p^nG$ to be the set, in fact it is a subgroup, of all $h\in G$ of the form $p^ng$ and define the p-height as the supremum of all $n$ such that $g \in p^nG$.
Now, one can continue and say $p^{\omega}G$ is the inersection of all the $p^nG$ and so on. 
Or formally, $p^0G = G$, $p^{\alpha+1}G= p(p^{\alpha}G)$ and $p^{\beta}G =\bigcap_{\alpha \lt \beta}$ for $\beta$ a limit ordinal.
This forms a descending chain of subgroups that (thus) eventually stabilzes; the ordinal where it stabilzes is called Ulm length. One can now also generalize the notion of p-height of g by defnining it as the ordinal $\sigma$ such that $g \in p^{\sigma}G$ yet not in $g \in p^{\sigma + 1}G$  and $\infty$ if such an ordinal does not exist.
And this notion is quite useful. For example, a classical result of Ulm (1930s) then says that two countable abelian $p$-groups (ie, order of each element a prime power)  $G$ and $H$ are isomophic if and only if $p^{\alpha}/p^{\alpha+1}G$ and $p^{\alpha}H/p^{\alpha+1}H$ are isomorphic for each $\alpha$ (these are called the Ulm factors) and the stable parts are isomorphic. 
And conversely for any ordinal one can construct a group of that Ulm length (and in addition one can prescribe to a considerable extent the Ulm factors at each point until there).