We know If a module satisfies both ACC and DCC then the largest chain is of finite length. Instead of defining largest chains we define maximal chains (which are more constructive and easier to work with) and show that all maximal chains have same length. Next, we construct a maximal chain of finite lenght and we are done. Now, I dont find this proof unnatural since, this is how we do things in other places like, to show finite dimensional vector spaces have same basis size, similarly in transcendence basis..we show one maximal has finite lenght and show all maximals have same length(matroidal property, also seen in trees in graphs)....
But I was trying for a more non constructive proof since i kind of made a guess first that this result should be true before finding out it is true, here is how i went about it:
I was trying to interpret what acc and dcc kind of meant. They both look different generalisations of finite dimensionality. Finite dimensional vector spaces are a special case. These can be defined as spaces in which largest chain of subspaces(largest in size) are also finite length. Now we if we ask this for modules then it holds: precisely;If the largest chain of modules for a module M is finite then M must satisfy both acc and dcc. Now the next question was, is the converse true? I find that it is and above i have described the standard proof found in Atiyah macdonald.
So I am trying to prove that if M satisfies both ACC and DCC then the length of the largest possible chain is of finite length.
Now obviously we cannot have chains of infinite length because of both ACC and DCC being satisfied. So, assume for sake of contradiction that we have chains of larger and larger lengths:
$M_{0}^{0}$
$M_{0}^{1} \subset M_{1}^{1} $
$M_{0}^{2} \subset M_{1}^{2} \subset M_{2}^{2}$
$\cdots$
Now look at the increasing chain formed by taking intersections columnwise $\cap_{i=0}^{\infty}M_{0}^{i} \subset \cap_{i=1}^{\infty}M_{1}^{i} \subset \cap_{i=2}^{\infty}M_{2}^{i}$
Now this chain must stabilise. Also each element of the chain is actually a finite intersection due to dcc. But I dont know how to use these to get a contradiction. Is there a way to extend this idea or a proof that doesn't involve proving all maximal chains have same length and so on. Please help. Any comments are welcome. I think we can learn something useful even if we dont find a non-constructive proof.